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ResearchCite this article Sutter D Fawzi O Renner R2016 Universal recovery map for approximateMarkov chains Proc R Soc A 472 20150623httpdxdoiorg101098rspa20150623

Received 8 September 2015Accepted 24 December 2015

Subject Areasquantum physics mathematical physicsquantum computing

Keywordsconditional mutual informationquantumMarkov chains recoverabilitystrong subadditivity

Author for correspondenceDavid Suttere-mail sutterditpphysethzch

Electronic supplementary material is availableat httpdxdoiorg101098rspa20150623 orvia httprsparoyalsocietypublishingorg

Universal recovery map forapproximate Markov chainsDavid Sutter1 Omar Fawzi23 and Renato Renner1

1Institute for Theoretical Physics ETH Zurich Zurich Switzerland2Department of Computing and Mathematical Sciences CaltechPasadena CA USA3LIP ENS de Lyon Lyon France

A central question in quantum information theoryis to determine how well lost information canbe reconstructed Crucially the correspondingrecovery operation should perform well withoutknowing the information to be reconstructed Inthis work we show that the quantum conditionalmutual information measures the performance ofsuch recovery operations More precisely we provethat the conditional mutual information I(A C|B)of a tripartite quantum state ρABC can be boundedfrom below by its distance to the closest recoveredstate RBrarrBC(ρAB) where the C-part is reconstructedfrom the B-part only and the recovery map RBrarrBC

merely depends on ρBC One particular applicationof this result implies the equivalence between twodifferent approaches to define topological order inquantum systems

1 IntroductionA state ρABC on a tripartite quantum system A otimes B otimes Cforms a (quantum) Markov chain if it can be recoveredfrom its marginal ρAB on A otimes B by a quantum operationRBrarrBC from B to B otimes C ie

ρABC =RBrarrBC(ρAB) (11)

An equivalent characterization of ρABC being a quantumMarkov chain is that the conditional mutual informationI(A C|B)ρ = H(AB)ρ + H(BC)ρ minus H(B)ρ minus H(ABC)ρ is zero[12] where H(A)ρ = minustr(ρA log2 ρA) is the von Neumannentropy The structure of these states has been studied invarious works In particular it has been shown that A andC can be viewed as independent conditioned on B for ameaningful notion of conditioning [3]

2016 The Authors Published by the Royal Society under the terms of theCreative Commons Attribution License httpcreativecommonsorglicensesby40 which permits unrestricted use provided the original author andsource are credited

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Very recently it has been shown that Markov states can be alternatively characterized byhaving a generalized Reacutenyi conditional mutual information that vanishes [4]

A natural question that is relevant for applications is whether the above statements are robust(In [5] an example is discussed that illustrates why this question is relevant In [6] furtherexplanations are given to emphasize the importance of this problem) Specifically one wouldlike to have a characterization of the states that have a small (but not necessarily vanishing)conditional mutual information ie I(A C|B) le ε for ε gt 0 First results revealed that such statescan have a large distance to Markov chains that is independent of ε [78] which has been takenas an indication that their characterization may be difficult However it has subsequently beenrealized that a more appropriate measure instead of the distance to a (perfect) Markov chain is toconsider how well (11) is satisfied [59ndash11] This motivated the definition of approximate Markovchains as states where (11) approximately holds

In recent work [6] it has been shown that the set of approximate Markov chains indeedcoincides with the set of states with small conditional mutual information In particular thedistance between the two terms in (11) which may be measured in terms of their fidelity F isbounded by the conditional mutual information1 More precisely for any state ρABC there exists atrace-preserving completely positive map RBrarrBC (the recovery map) such that

I(A C|B)ρ ge minus2 log2 F(ρABCRBrarrBC(ρAB)) (12)

Furthermore a converse inequality of the form I(A C|B)2ρ le minusc2 log2 F(ρABCRBrarrBC(ρAB)) where

c depends logarithmically on the dimension of A can be shown to hold [611]We also note that the fidelity term in (12) maximized over all recovery maps ie

F(A C|B)ρ = supRBrarrBC

F(ρABCRBrarrBC(ρAB)) (13)

is called fidelity of recovery2 and has been introduced and studied in [1415] With this quantity themain result of [6] can be written as

I(A C|B)ρ ge minus2 log2 F(A C|B)ρ (14)

The fidelity of recovery has several natural properties eg it is monotonous under localoperations on A and C and it is multiplicative [15]

The result of [6] has been extended in various ways Based on quantum state redistributionprotocols it has been shown in [16] that (12) still holds if the fidelity term is replaced by themeasured relative entropy DM(middot middot) which is generally larger ie there exists a recovery map RBrarrBC

such that

I(A C|B)ρ ge DM(ρABCRBrarrBC(ρAB)) ge minus2 log2 F(ρABCRBrarrBC(ρAB)) (15)

The measured relative entropy is defined as the supremum of the relative entropy with measuredinputs over all projective measurements3 M= Mx ie

DM(ρσ ) = sup

D(M(ρ)M(σ )) M(ρ) =

sumx

tr(ρMx)|x〉〈x| withsum

xMx = id

(16)

where |x〉 is a finite set of orthonormal vectors This quantity was studied in [1718]Furthermore in [15] an alternative proof of (12) has been derived that uses properties of the

fidelity of recovery (in particular multiplicativity) Another recent work [19] showed how togeneralize ideas from [6] to prove a remainder term for the monotonicity of the relative entropyin terms of a recovery map that satisfies (12)

1The fidelity of ρ and σ is defined as F(ρ σ ) = radicρradicσ1

2We note that if A B and C are finite-dimensional Hilbert spaces the supremum is achieved since the set of recovery maps iscompact (see remark 103 in the electronic supplementary material) and the fidelity is continuous in the input state (see [1213]or lemma B9 in [6])3Without loss of generality these can be assumed to be rank-one projectors

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3


All known proofs of (12) are non-constructive in the sense that the recovery map RBrarrBC isnot given explicitly It is merely known [6] that if A B and C are finite-dimensional then RBrarrBC

can always be chosen such that it has the form

XB rarr VBCρ12BC

(ρ

minus12B UBXBUdagger

Bρminus12B otimes idC

)ρ

12BC Vdagger

BC (17)

on the support of ρB where UB and VBC are unitaries on B and B otimes C respectively It would benatural to expect that the choice of the recovery map that satisfies (12) only depends on ρBChowever this is only known in special cases One such special case is Markov chains ρABC iestates for which (11) holds perfectly Here a map of the form (17) with VBC = idBC and UB = idB

(sometimes referred to as transpose map or Petz recovery map) serves as a perfect recovery map[12] Another case where a recovery map that only depends on ρBC is known explicitly are stateswith a classical B system ie qcq-states of the form ρABC =sum

b PB(b)|b〉〈b| otimes ρACb where PB is aprobability distribution |b〉b an orthonormal basis on B and ρACbb a family of states on A otimes CAs discussed in [6] for such states (12) holds for the recovery map defined by RBrarrBC(|b〉〈b|) =|b〉〈b| otimes ρCb for all b where ρCb = trA(ρACb) For general states however the previous results leftopen the possibility that the recovery map RBrarrBC depends on the full state ρABC rather than themarginal ρBC only In particular the unitaries UB and VBC in (17) although acting only on Brespectively B otimes C could have such a dependence

In this work we show that for any state ρBC on B otimes C there exists a recovery map RBrarrBC thatis universalmdashin the sense that the distance between any extension ρABC of ρBC and RBrarrBC(ρAB) isbounded from above by the conditional mutual information I(A C|B)ρ In other words we showthat (12) remains valid if the recovery map is chosen depending on ρBC only rather than on ρABCThis result implies a close connection between two different approaches to define topologicalorder of quantum systems

2 Main resultTheorem 21 For any density operator ρBC on B otimes C there exists a trace-preserving completely

positive map RBrarrBC such that for any extension ρABC on A otimes B otimes C


where A B and C are separable Hilbert spaces

Remark 22 If B and C are finite-dimensional Hilbert spaces the statement of theorem 21 canbe tightened to

I(A C|B)ρ ge DM(ρABC RBrarrBC(ρAB)) (22)

Remark 23 The recovery map RBrarrBC predicted by theorem 21 has the property that itmaps ρB to ρBC To see this note that I(A C|B)ρ = 0 for any density operator of the formρABC = ρA otimes ρBC Theorem 21 thus asserts that ρABC must be equal to RBrarrBC(ρAB) whichimplies that ρBC =RBrarrBC(ρB) We note that so far it was unknown whether recovery maps thatsatisfy (12) and have this property do exist

We note that theorem 21 does not reveal any information about the structure of the recoverymap that satisfies (21) However if we consider a linearized version of the bound (21) we canmake more specific statements

Corollary 24 For any density operator ρBC on B otimes C there exists a trace-preserving completelypositive map RBrarrBC such that for any extension ρABC on A otimes B otimes C

I(A C|B)ρ ge 2ln(2)

(1 minus F(ρABCRBrarrBC(ρAB))

)(23)


4


where A B and C are separable Hilbert spaces Furthermore if B and C are finite-dimensional then RBrarrBC

has the formXB rarr ρ

12BC UBCrarrBC

(ρ

minus12B XBρ

minus12B otimes idC

)ρ

12BC (24)

on the support of ρB where UBCrarrBC is a unital trace-preserving map from B otimes C to B otimes C

Remark 25 Following the proof of corollary 24 we can deduce a more specific structure ofthe universal recovery map In the finite-dimensional case the map RBrarrBC satisfying (23) can beassumed to have the form

XB rarrint

VsBCρ

12BC(ρ

minus12B Us

BXBUsdaggerB ρ

minus12B otimes idC

)ρ

12BC Vsdagger

BCμ(ds) (25)

where μ is a probability measure on some set S VsBCsisinS is a family of unitaries on B otimes C

that commute with ρBC and UsBsisinS is a family of unitaries on B that commute with ρB

However the representation of the recovery map given in (24) has certain advantages comparedto the representation (25) The fidelity maximized over all recovery maps of the form (24) canbe phrased as a semidefinite programme and therefore be computed efficiently whereas it isunknown whether the same is possible for (25)

We note that for almost all density operators ρBC ie for all ρBC except for a set of measurezero we can replace the unitaries Us

B and VsBC by complex matrix exponentials of the form ρit

B andρit

BC respectively with t isin R This shows that (25) without the integral (the integration in (25) isonly necessary to guarantee that the recovery map is universal) coincides with the recovery mapfound in [20]4

Example 26 For density operators with a marginal on B otimes C of the form ρBC = ρB otimes ρC auniversal recovery map that satisfies (22) is uniquely defined on the support of ρBmdashit is thetranspose map which in this case simplifies to RBrarrBC XB rarr XB otimes ρC It is straightforward tosee that (22) holds In fact we even have equality if we consider the relative entropy (which is ingeneral larger than the measured relative entropy) ie

I(A C|B)ρ = D(ρABCRBrarrBC(ρAB)

) (26)

The uniqueness of RBrarrBC on the support of ρB follows by using the fact that the universalrecovery map should perfectly recover the Markov state ρAB otimes ρC where ρAB is a purificationof ρB This forces RBrarrBC to agree with the transpose map on the support of ρB [12]

The proof of theorem 21 is structured into two parts We first prove the statement for finite-dimensional Hilbert spaces B and C in sect4 and then show that this implies the statement forgeneral separable Hilbert spaces in sect5 The proof of corollary 24 is given in sect6

3 ApplicationsA celebrated result known as strong subadditivity states that the conditional quantum mutualinformation of any density operator is non-negative [2324] ie

I(A C|B)ρ ge 0 (31)

for any density operator ρABC on A otimes B otimes C Theorem 21 implies a strengthened version ofthis inequality with a remainder term that is universal in the sense that it only depends onρBC The conditional quantum mutual information is a useful tool in different areas of physicsand computer science It is helpful to characterize measures of entanglement [625] analyse thecorrelations of quantum many-body systems [526] prove quantum de Finetti results [2728] andmake statements about quantum information complexity [29ndash31] It is expected that oftentimeswhen (12) can be used its universal version (predicted by theorem 21) is even more helpful

In the following we sketch an application where the universality result is indispensableTheorem 21 can be applied to establish a connection between two alternative definitions of

4This follows by the equidistribution theorem which is a special case of the strong ergodic theorem [21 sectII5] (see also [22])


5


A C

B

B

Figure 1 Relevant topology of the subsystems A B and C such that a stateρABC exhibits TQOprime if I(A C|B)ρ = 2γ gt 0

topological order of quantum systems (denoted by TQO and TQOprime) Consider an n-spin system withn isin N While the following statements should be understood asymptotically (in the limit n rarr infin)we omit the dependence on n in our notation for simplicity

According to [32] a family of states ρiiisinI with ρi isin E for all i isin I and |I|ltinfin where E denotesa collection of states exhibits topological quantum order (TQO) if and only if any two membersof the family

(i) are (asymptotically) orthogonal ie F(ρi ρj) = 0 for all i = j isin I and(ii) have (asymptotically) the same marginals on any sufficiently small subregion ie trGρ

i =trGρ

j for all i j isin I and G sufficiently large5

Alternatively for three regions A B and C that form a certain topology F (see figure 1 and [33]) astate ρABC on such a subspace exhibits topological quantum order (TQOprime) if I(A C|B)ρ = 2γ gt 0where γ denotes a topological entanglement entropy [33]6 (See [33] for more explanations on howthe topological entanglement entropy is defined for the topology F depicted in figure 1)

It is an open problem to find out how these two characterizations are related eg if afamily K of states on F that exhibits TQO implies that most of its members have TQOprime Thisconnection follows by theorem 21 Suppose ρiiisinI with ρi isinK for all i isin I shows TQO Thenconsider subsystems A B and C that together form a non-contractible loop By definition ofTQO the density operators ρiiisinI share (asymptotically) the same marginals on B otimes C Applyingtheorem 21 to this common marginal together with the continuity of the conditional mutualinformation [35] ensures that there exist a recovery map RBrarrBC such that for any i isin I

I(A C|B)ρi ge minus2 log F(ρiABCRBrarrBC(ρi

AB)) (32)

Since the density operators ρiABCiisinI are (asymptotically) orthogonal share (asymptotically) the

same marginals on A otimes B and the fidelity is continuous in its inputs [1213] this implies that forall i isin I except of a single element we have

I(A C|B)ρi ge constgt 0 (33)

4 Proof for finite dimensionsThroughout this section we assume that the Hilbert spaces B and C are finite-dimensional In theproof Steps 1ndash3 below we also make the same assumption for A but then drop it in Step 4 Westart by explaining why (22) is a tightened version of (21) which was noticed in [16] Let Dα(middotmiddot)be the α-Quantum Reacutenyi Divergence as defined in [3637] with D1(ρσ ) = D(ρσ ) = tr(ρ(log ρ minuslog σ )) and Dα(ρσ ) = (1(α minus 1)) log tr((σ (1minusα)2αρσ (1minusα)2α)α) for any density operator ρ anynon-negative operator σ such that supp(ρ) sube supp(σ ) and any α isin (0 1) cup (1 infin) By definition of

5More precisely we require that trG(ρi) minus trG(ρj)1 = o(nminus2)

6Note that there exist different quantities that are called topological entanglement entropy (see also [34])


6


the measured relative entropy (see (16)) we find for any two states ρ and σ

DM(ρσ ) = supMisinM

D(M(ρ)M(σ )) ge supMisinM

D12(M(ρ)M(σ ))

= minus2 log2 infMisinM

F(M(ρ)M(σ )) = minus2 log2 F(ρ σ ) (41)

where M = M M(ρ) =sumx tr(ρMx)|x〉〈x| with

sumx Mx = id and |x〉 is a family of orthonormal

vectors The inequality step uses that α rarr Dα(ρσ ) is a monotonically non-decreasing function inα [36 theorem 7] and the final step follows from the fact that for any two states there exists anoptimal measurement that does not increase their fidelity [38 sect33] As a result in order to provetheorem 21 for finite-dimensional B and C it suffices to prove (22)

We first derive a proposition (proposition 41) and then show how it can be used to prove (22)(and hence theorem 21) The proposition refers to a family of functions

D(A otimes B otimes C) ρ rarrR(ρ) isin R cup minusinfin (42)

parametrized by recovery maps R isin TPCP(B B otimes C) where TPCP(B B otimes C) denotes the set oftrace-preserving completely positive maps from B to B otimes C and D(A otimes B otimes C) denotes the setof density operators on A otimes B otimes C Subsequently in the proof the function family R(middot) will beconstructed as the difference of the two terms in (22) (see equation (438)) such that R(ρ) ge 0corresponds to (22) The proposition asserts that if for any extension ρABC of ρBC we haveR(ρ) ge 0 for some R isin TPCP(B B otimes C) and provided the function family R(middot) satisfies certainproperties described below then there exists a single recovery map R for which R(ρ) ge 0 for allextensions ρABC of ρBC on a fixed A system We note that the precise form of the function familyR(middot) is irrelevant for proposition 41 as long as it satisfies a list of properties as stated below

As described above our goal is to prove that there exists a recovery map RBrarrBC suchthat R(ρ) ge 0 for all ρABC isin D(A otimes B otimes C) with a fixed marginal ρBC on B otimes C To formulateour argument more concisely we introduce some notation For any set S of density operatorsρABC isin D(A otimes B otimes C) we define

R(S) = infρisinS

R(ρ) (43)

The desired statement then reads as R(S) ge 0 for any set S of states on A otimes B otimes C with a fixedmarginal ρBC Furthermore for any fixed states ρ0

ABC and ρABC on A otimes B otimes C and p isin [0 1] wedefine

ρp

AABC= (1 minus p)|0〉〈0|A otimes ρ0

ABC + p|1〉〈1|A otimes ρABC (44)

where A is an additional system with two orthogonal states |0〉 and |1〉 More generally for anyfixed state ρ0

ABC and for any set S of density operators ρABC we set

Sp =ρ

p

AABC ρABC isin S

(45)

Required properties of the -function 1

(i) For any ρ0ABC ρABC isin D(A otimes B otimes C) with identical marginals ρ0

BC = ρBC on B otimes C for anyR isin TPCP(B B otimes C) and for any p isin [0 1] we have R(ρp) = (1 minus p)R(ρ0) + pR(ρ)

(ii) For any RRprime isin TPCP(B B otimes C) for any α isin [0 1] and R= αR + (1 minus α)Rprime we haveR(ρ) ge αR(ρ) + (1 minus α)Rprime (ρ) for all ρ isin D(A otimes B otimes C)

(iii) For any R isin TPCP(B B otimes C) the function D(A otimes B otimes C) ρ rarrR(ρ) isin R cup minusinfin isupper semicontinuous

(iv) For any ρ isin D(A otimes B otimes C) the function TPCP(B B otimes C) R rarrR(ρ) isin R cup minusinfin isupper semicontinuous


7


Property (i) implies that for any state ρ0ABC for any set S of operators ρABC with ρBC = ρ0

BC andfor any p isin [0 1] we have

R(Sp) = infρisinS

R(ρp) = (1 minus p)R(ρ0) + p infρisinS

R(ρ) = (1 minus p)R(ρ0) + pR(S) (46)

Similarly property (ii) implies

R(S) = infρisinS

R(ρ) ge infρisinS

αR(ρ) + (1 minus α)Rprime (ρ)

ge α infρisinS

R(ρ) + (1 minus α) infρisinS

Rprime (ρ) = αR(S) + (1 minus α)Rprime (S) (47)

Proposition 41 Let A B and C be finite-dimensional Hilbert spaces P sube TPCP(B B otimes C) be compactand convex S be a set of density operators on A otimes B otimes C with identical marginals on B otimes C andR(middot) bea family of functions of the form (42) that satisfies properties (i)ndash(iv) Then

forallρ isin SexistR isinP R(ρ) ge 0 rArr existR isinP R(S) ge 0 (48)

We now proceed in four steps In the first we prove proposition 41 for finite sets S This is doneby induction over the cardinality of the set S We show that if the statement of proposition 41 istrue for all sets S with |S| = n this implies that it remains valid for all sets S with |S| = n + 1In Step 2 we use an approximation step to extend this to infinite sets S which then completesthe proof of proposition 41 In the final two steps we show how to conclude the statement oftheorem 21 for the finite-dimensional case from that In Step 3 we prove (22) for the case wherethe recovery map that satisfies (22) could still depend on the dimension of the system A In Step 4we show how this dependency can be removed

Proposition 41 resembles Sionrsquos minimax theorem [39] After the completion of this work ithas been noticed that the argument done by proposition 41 in this work can be alternativelycarried out using Sionrsquos minimax theorem (see [40] for a detailed explanation)

(a) Step 1 Proof of proposition 41 for finite size setsSWe proceed by induction over the cardinality n = |S| of the set S of density operators Moreprecisely the induction hypothesis is that for any finite-dimensional Hilbert space A and anyset S of size n consisting of density operators on A otimes B otimes C with fixed marginal ρBC on B otimes Cthe statement (48) holds For n = 1 this hypothesis holds trivially for R=R7

We now prove the induction step Suppose that the induction hypothesis holds for some n LetA be a finite-dimensional Hilbert space and let S cup ρ0

ABC be a set of cardinality n + 1 where S isa set of states on A otimes B otimes C with fixed marginal ρBC on B otimes C of cardinality n and ρ0

ABC is anotherstate with ρ0

BC = ρBC We need to prove that there exists a recovery map RBrarrBC isinP such that

R(S cup ρ0ABC) ge 0 (49)

Let p isin [0 1] and consider the set Sp as defined in (45) In the following we view the statesρp (see equation (44)) in this set as tripartite states on (A otimes A) otimes B otimes C ie we regard the systemA otimes A as one (larger) system The induction hypothesis applied to the extension space A otimes A andthe set Sp (of size n) of states on (A otimes A) otimes B otimes C implies the existence of a map Rp

BrarrBC isinP suchthat

Rp (Sp) ge 0 (410)

As by assumption the function D(A otimes B otimes C) ρ rarrRp (ρ) isin R cup minusinfin satisfies property (i) (andhence also equation (46)) we obtain

(1 minus p)Rp (ρ0) + pRp (S) ge 0 (411)

7For n = 0 we have R(S) = infin ge 0 for any R isinP since the infimum of an empty set is infinity


8


This implies that

Rp (ρ0) ge 0 or Rp (S) ge 0 (412)

Furthermore for p = 0 the left inequality holds and for p = 1 the right inequality holds Bychoosing K0 = p isin [0 1] Rp (ρ0) ge 0 and K1 = p isin [0 1] Rp (S) ge 0 the touching sets lemma(see lemma 111 in the electronic supplementary material) implies that for any δ gt 0 there existu v isin [0 1] with 0 le v minus u le δ such that

Ru (ρ0) ge 0 and Rv (S) ge 0 (413)

Note also that RuBrarrBC Rv

BrarrBC isinP as by the induction hypothesis RpBrarrBC isinP for any p isin [0 1]

We will use this to prove that the recovery map R isinP defined by

R = αRu + (1 minus α)Rv (414)

for an appropriately chosen α isin [0 1] satisfies

R(ρ0) ge minuscδ and R(S) ge minuscδ (415)

where c is a constant defined by

c = 4 maxRisinTPCP(BBotimesC)

maxρisinD(AotimesBotimesC)

R(ρ)ltinfin (416)

Properties (iii) and (iv) together with simple topological facts about the set of density operatorsand the set of trace-preserving completely positive maps (see lemma 101 and remark 103 statedin the electronic supplementary material) ensure that the two maxima in (416) are attained whichimplies by the definition of the codomain of R(middot) (see equation (42)) that c is finite In otherwords for any δ gt 0 there exists a recovery map Rδ isinP such that

Rδ (S cup ρ0) ge minuscδ (417)

The compactness of P ensures that there exists a recovery map R isinP and a sequence δnnisinN suchthat

limnrarrinfin δn = 0 and lim

nrarrinfin Rδn = R (418)

Because of (417) we have

lim supnrarrinfin

Rδn (S cup ρ0) ge limnrarrinfin minuscδn = 0 (419)

which together with property (iv) implies that

R(S cup ρ0) = minρisinScupρ0

R(ρ) ge minρisinScupρ0

lim supnrarrinfin

Rδn (ρ) ge lim supnrarrinfin

minρisinScupρ0

Rδn (ρ)

= lim supnrarrinfin

Rδn (S cup ρ0) ge 0 (420)

and thus proves (49)It thus remains to show (415) To simplify the notation let us define

Λ0 =Ru (ρ0) and Λ1 =Rv (S) (421)

as well as

Λ0 =Rv (ρ0) and Λ1 =Ru (S) (422)

It follows from (411) that

(1 minus u)Λ0 + uΛ1 ge 0 (423)

Similarly we have

(1 minus v)Λ0 + vΛ1 ge 0 (424)


9


As by assumption the function R(middot) satisfies property (ii) we find together with (424) that forany α isin [0 1] and R= αRu + (1 minus α)Rv

R(ρ0) ge αRu (ρ0) + (1 minus α)Rv (ρ0) = αΛ0 + (1 minus α)Λ0 ge αΛ0 minus (1 minus α)v

1 minus vΛ1 (425)

(If v = 1 it suffices to consider the case α= 1 so that the last term can be omittedcf equation (429)) Analogously using (47) and (423) we find

R(S) ge αRu (S) + (1 minus α)Rv (S) = αΛ1 + (1 minus α)Λ1 ge minusα 1 minus uu

Λ0 + (1 minus α)Λ1 (426)

(If u = 0 it suffices to consider the case α = 0 cf equation (432))To conclude the proof of (415) it suffices to choose α isin [0 1] such that the terms on the right-

hand side of (425) and (426) satisfy

αΛ0 minus (1 minus α)v

1 minus vΛ1 ge minuscδ (427)

and

minus α1 minus u

uΛ0 + (1 minus α)Λ1 ge minuscδ (428)

Let us first assume that u ge 12 Since Λ0 and Λ1 are non-negative (see equation (413)) we may

choose α isin [0 1] such thatα(1 minus v)Λ0 = (1 minus α)vΛ1 (429)

This immediately implies that the left-hand side of (427) equals 0 so that the inequality holdsAs 1

2 le u le v le 1 and v minus u le δ we have∣∣∣∣1 minus uu

minus 1 minus v

v

∣∣∣∣le 4δ (430)

Combining this with (429) we find

minus α1 minus u

uΛ0 + (1 minus α)Λ1 ge minusαΛ0

(1 minus v

v+ 4δ

)+ (1 minus α)Λ1 = minus4αΛ0δ ge minus4Λ0δ (431)

which proves (428) because by (416) we have Λ0 le c4Analogously if ult 1

2 choose α isin [0 1] such that

α(1 minus u)Λ0 = (1 minus α)uΛ1 (432)

This immediately implies that the left-hand side of (428) equals 0 so that the inequality holdsFurthermore for δ gt 0 sufficiently small such that v le 1

2 we obtain∣∣∣∣ v

1 minus vminus u

1 minus u

∣∣∣∣lt 4δ (433)

Together with (432) this implies


1 minus vΛ1 ge αΛ0 minus (1 minus α)Λ1

(u

1 minus u+ 4δ

)= minus4(1 minus α)Λ1δ ge minus4Λ1δ (434)

which establishes (427) This concludes the proof of proposition 41 for sets S of finite size

(b) Step 2 Extension to infinite setsSAll that remains to be done to prove proposition 41 is to generalize the statement to arbitrarilylarge sets S In fact we show that there exists a recovery map RBrarrBC isinP such that R(S) ge 0where S is the set of all density operators on A otimes B otimes C for a fixed finite-dimensional Hilbertspace A and a fixed marginal ρBC

Note first that this set S of all density operators on A otimes B otimes C with fixed marginal ρBC on B otimes Cis compact (see lemma 102 in the electronic supplementary material) This implies that for anyε gt 0 there exists a finite set Sε of density operators on A otimes B otimes C such that any ρ isin S is ε-close


10


to an element of Sε We further assume without loss of generality that Sεprime sub Sε for εprime ge ε LetRε isin TPCP(B B otimes C) be a map such that Rε (Sε) ge 0 whose existence follows from the validityof proposition 41 for sets of finite size (which we proved in Step 1) Since the set TPCP(B B otimes C)is compact (see remark 103 in the electronic supplementary material) there exists a decreasingsequence εnnisinN and R isin TPCP(B B otimes C) such that

limnrarrinfin εn = 0 and R= lim

nrarrinfinRεn (435)

Combining this with property (iv) gives for all n isin N

R(Sεn ) = infρisinSεn

R(ρ) ge infρisinSεn

lim supmrarrinfin

Rεm (ρ) ge lim supmrarrinfin

infρisinSεn

Rεm (ρ)

ge lim supmrarrinfin

infρisinSεm

Rεm (ρ) = lim supmrarrinfin

Rεm (Sεm ) ge 0 (436)

where the third inequality holds since Sεn sub Sεm for εn ge εm respectively n le m The finalinequality follows from the defining property of Rε For any fixed ρ isin S and for all n isin N letρn isin Sεn be such that limnrarrinfin ρn = ρ isin S (By definition of Sεn it follows that such a sequenceρnnisinN with ρn isin Sεn always exists) Property (iii) together with (436) yields

R(ρ) =R(

limnrarrinfin ρn

)ge lim sup

nrarrinfinR(ρn) ge lim sup

nrarrinfinR(Sεn ) ge 0 (437)

Since (437) holds for any ρ isin S we obtain R(S) ge 0 which completes the proof ofproposition 41

(c) Step 3 From proposition 41 to theorem 21 for fixed system AWe next show that theorem 21 for the case where A is a fixed finite-dimensional system followsfrom proposition 41 For this we use proposition 41 for the function family

R D(A otimes B otimes C) rarr R cup minusinfinρABC rarr I(A C|B)ρ minus DM(ρABCRBrarrBC(ρAB)) (438)

with RBrarrBC isin TPCP(B B otimes C) We note that since C is finite-dimensional this implies thatR(ρ)ltinfin for all ρ isin D(A otimes B otimes C) To apply proposition 41 we have to verify that the functionfamily D(A otimes B otimes C) ρ rarrR(ρ) isin R cup minusinfin of the form (438) satisfies the assumptions of theproposition This is ensured by the following lemma

Lemma 42 Let A be a separable and B and C finite-dimensional Hilbert spaces The function familyR(middot) defined by (438) satisfies properties (i)ndash(iv)

Proof We first verify that the function R(middot) satisfies property (i) For any state ρp of theform (44) we have by the definition of the mutual information

I(AA C|B)ρp = H(C|B)ρp minus H(C|BAA)ρp (439)

Because ρ0BC = ρBC the first term H(C|B)ρp is independent of p ie H(C|B)ρp = H(C|B)ρ0 =

H(C|B)ρ The second term can be written as an expectation over A ie

H(C|BAA)ρp = (1 minus p)H(C|BA)ρ0 + pH(C|BA)ρ (440)

As a result we find

I(AA C|B)ρp = (1 minus p)I(A C|B)ρ0 + pI(A C|B)ρ (441)


11


The density operator RBrarrBC(ρp

AAB) can be written as

RBrarrBC(ρp

AAB) = (1 minus p)|0〉〈0|A otimes RBrarrBC(ρ0

AB) + p|1〉〈1|A otimes RBrarrBC(ρAB) (442)

We can thus apply lemma 93 given in the electronic supplementary material (which states alinearity property of the measured relative entropy for orthogonal states) from which we obtain

DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)

Equations (441) and (443) imply that

R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)

which concludes the proof of property (i)That R(middot) satisfies property (ii) can be seen as follows Let RBrarrBCRprime

BrarrBC isin TPCP(B B otimes C)α isin [0 1] and RBrarrBC = αRBrarrBC + (1 minus α)Rprime

BrarrBC Since the measured relative entropy is convexin the second argument (see lemma 94 given in the electronic supplementary material) we findthat for any state ρABC on A otimes B otimes C

DM(ρABCRBrarrBC(ρAB)) = DM

(ρABCαRBrarrBC(ρAB) + (1 minus α)Rprime

BrarrBC(ρAB))

le αDM(ρABCRBrarrBC(ρAB)) + (1 minus α)DM(ρABCRprimeBrarrBC(ρAB)) (445)

and henceR(ρ) ge αR(ρ) + (1 minus α)Rprime (ρ) (446)

We next verify that the function R(middot) satisfies property (iii) The AlickindashFannesinequality ensures that D(A otimes B otimes C) ρ rarr I(A C|B)ρ isin R

+ is continuous since C is finite-dimensional [35] By the definition of R(middot) it thus suffices to show that D(A otimes B otimes C) ρABC rarrDM(ρABCRBrarrBC(ρAB)) isin R

+ is lower semicontinuous Let ρnABCnisinN be a sequence of states on

A otimes B otimes C such that limnrarrinfin ρnABC = ρABC isin D(A otimes B otimes C) By definition of the measured relative

entropy (see (16)) we find for M = M M(ρ) =sumx tr(ρMx)|x〉〈x| with

sumx Mx = id

lim infnrarrinfin DM(ρn

ABCRBrarrBC(ρnAB)) = lim inf

nrarrinfin supMisinM

D(M(ρnABC)M(RBrarrBC(ρn

AB)))

ge supMisinM

lim infnrarrinfin D(M(ρn

ABC)M(RBrarrBC(ρnAB)))

ge supMisinM

D(M(ρABC)M(RBrarrBC(ρAB)))

= DM(ρABCRBrarrBC(ρAB)) (447)

In the penultimate step we use that the relative entropy is lower semicontinuous [41Exercise 722] and that M as well as RBrarrBC are linear and bounded operators and hencecontinuous

We finally show that R(middot) fulfils property (iv) It suffices to verify that TPCP(B B otimes C) R rarrDM(ρABCR(ρAB)) isin R

+ is lower semicontinuous where by definition of the measured relativeentropy (see (16)) we have that

DM(ρABCR(ρAB)) = supMisinM

D(M(ρABC)M(RBrarrBC(ρAB))) (448)

Note that since R and M are linear bounded operators and hence continuous and the relativeentropy for two states σ1 and σ2 is defined by D(σ1σ2) = tr(σ1(log σ1 minus log σ2)) we find that R rarrD(M(ρABC)M(RBrarrBC(ρAB))) is continuous as the logarithm R

+ x rarr log x isin R is continuousSince the supremum of continuous functions is lower semicontinuous [42 ch IV Section 62Theorem 4] the assertion follows

What remains to be shown in order to apply proposition 41 is that for any ρ isin S where Sis the set of states on A otimes B otimes C with a fixed marginal ρBC on B otimes C there exists a recovery


12


map RBrarrBC isinP such that R(ρ) ge 0 By choosing P = TPCP(B B otimes C) the main result of [16]however precisely proves this We have thus shown thatR(ρ) ge 0 holds for a universal recoverymap RBrarrBC isinP so that (22) follows for any fixed dimension of the A system This proves thestatement of remark 22 (and hence theorem 21) for the case where A is a fixed finite-dimensionalHilbert space

(d) Step 4 Independence from the A systemLet S be the set of all density operators on A otimes B otimes C with a fixed marginal ρBC on B otimes C whereB and C are finite-dimensional Hilbert spaces and A is the infinite-dimensional Hilbert space 2

of square summable sequences We now show that there exists a recovery map RBrarrBC such thatR(S) ge 0

Let ΠaAaisinN be a sequence of finite-rank projectors on A that converges to idA with respect to

the weak operator topology Let Sa denote the set of states whose marginal on A is contained inthe support of Πa

Aand with the same fixed marginal ρBC on B otimes C as the elements of S For all

a isin N let RaBrarrBC denote a recovery map that satisfies Ra (Sa) ge 0 Note that the existence of such

maps is already established by the proof of theorem 21 for the finite-dimensional case As theset of trace-preserving completely positive maps on finite-dimensional systems is compact (seeremark 103 in the electronic supplementary material) there exists a subsequence aiiisinN such thatlimirarrinfin ai = infin and limirarrinfin Rai = R isin TPCP(B B otimes C) For every ρ isin S there exists a sequence ofstates ρaaisinN with ρa isin Sa that converges to ρ in the trace norm (see lemma 123 in the electronicsupplementary material) Lemma 42 (in particular properties (iii) and (iv)) yields for any ρ isin S

R(ρ) ge lim supararrinfin

R(ρa) ge lim supararrinfin

lim supirarrinfin

Rai (ρa) ge lim supararrinfin

lim supirarrinfin

infρisinSa

Rai (ρ)

ge lim supirarrinfin

infρisinSai

Rai (ρ) = lim supirarrinfin

Rai (Sai ) ge 0 (449)

The fourth inequality follows since ai ge a for large enough i and since this implies that Sai sup Saand the final inequality follows by definition of Rai This shows that R(S) ge 0

To retrieve the statement of remark 22 (and hence theorem 21 for finite-dimensional B and C)we need to argue that this same map R remains valid when we consider any separable space AIn order to do this observe that any separable Hilbert space A can be isometrically embeddedinto A [21 Theorem II7] To conclude it suffices to remark that R is invariant under isometriesapplied on the space A

5 Extension to infinite dimensionsIn this section we show how to obtain the statement of theorem 21 for separable (not necessarilyfinite-dimensional) Hilbert spaces A B C from the finite-dimensional case that has been provenin sect4 For trace non-increasing completely positive maps RBrarrBC we define the function family

R D(A otimes B otimes C) rarr R cup minusinfinρABC rarr F(ρABCRBrarrBC(ρAB)) minus 2minus(12)I(AC|B)ρ (51)

where D(A otimes B otimes C) denotes the set of states on A otimes B otimes C We will use the same notation asintroduced at the beginning of sect4 In addition we take S to be the set of all states on A otimes B otimes Cwith a fixed marginal ρBC on B otimes C The proof proceeds in two steps where we first show thatthere exists a sequence of recovery maps Rk

BrarrBCkisinN such that limkrarrinfin Rk (S) ge 0 where theproperty that all elements of S have the same marginal on the B otimes C system will be importantIn the second step we conclude by an approximation argument that there exists a recovery mapRBrarrBC such that R(S) ge 0


13


(a) Step 1 Existence of a sequence of recovery mapsWe start by introducing some notation that is used within this step Let Πb

BbisinN and Π cCcisinN

be sequences of finite-rank projectors on B and C which converge to idB and idC with respectto the weak operator topology For any given ρABC isin D(A otimes B otimes C) consider the normalizedprojected states

ρbcABC = (idA otimesΠb

B otimesΠ cC)ρABC(idA otimesΠb

B otimesΠ cC)

tr((idA otimesΠbB otimesΠ c

C)ρABC)(52)

and

ρcABC = (idA otimes idB otimesΠ c

C)ρABC(idA otimes idB otimesΠ cC)

tr((idA otimes idB otimesΠ cC)ρABC)

(53)

where for any c isin N the sequence ρbcABCbisinN converges to ρc

ABC in the trace norm (see corollary 2of [43] or lemma 121 in the electronic supplementary material) and the sequence ρc

ABCcisinN

converges to ρABC also in the trace norm Let Sbc be the set of states that is generated by (52) for allρABC isin S We note that for any given b c all elements of Sbc have an identical marginal on B otimes CLet Rbc

BrarrBC denote a recovery map that satisfies Rbc (Sbc) ge 0 whose existence is established inthe proof of theorem 21 for finite-dimensional systems B and C (see sect4) We next state a lemmathat explains how R(ρ) changes when we replace ρ by a projected state ρbc

Lemma 51 For any ρBC isin D(B otimes C) there exists a sequence of reals ξ bcbcisinN with8

limcrarrinfin limbrarrinfin ξ bc = 0 such that for any R isin TPCP(B B otimes C) any extension ρABC of ρBC and ρbcABC

as given in (52) we have

R(ρbc) minus R(ρ) le ξ bc for all b c isin N (54)

Proof We note that local projections applied to the subsystem C can only decrease the mutualinformation ie

tr(Π cCρC)I(A C|B)ρc le I(A C|B)ρ (55)

To see this assume that a measurement with respect to Π cC as well as its orthogonal complement

is applied to ρ Furthermore let Z be a random variable that stores the outcome of thismeasurement Then by the data processing inequality

I(A C|B)ρ = H(A|B)ρ minus H(A|CB)ρ ge H(A|B)ρ prime minus H(A|CBZ)ρ prime

ge H(A|BZ)ρ prime minus H(A|CBZ)ρ prime = I(A C|BZ)ρ prime (56)

where ρprime is the state after the measurement Because I(A C|BZ)ρ prime can be written as the expectationover the mutual information of the post-measurement states conditioned on the different valuesof Z and because all these terms are non-negative the above claim follows

The AlickindashFannes inequality [35] ensures that for a fixed finite-dimensional system C theconditional mutual information I(A C|B)ρ = H(C|B)ρ minus H(C|AB)ρ is continuous in ρ with respectto the trace norm ie

I(A C|B)ρbc le I(A C|B)ρc + 8εbc log(rankΠ c

C)+ 4h(εbc) (57)

where εbc = ∥∥ρbcABC minus ρc

ABC

∥∥1 and h(middot) denotes the binary Shannon entropy function defined by

h(p) = minusp log2(p) minus (1 minus p) log2(1 minus p) for 0 le p le 1 Using the Fuchsndashvan de Graaf inequality [44]

8The precise form of the sequence ξ bcbcisinN is given in the proof (see equation (517))


14


and a variant of the gentle-measurement lemma (see lemma 121 given in the electronicsupplementary material) we find

εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2

radicradicradicradic1 minus

tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)

Combining (55) and (57) yields

I(A C|B)ρbc le 1tr(Π c

CρC)I(A C|B)ρ + 8εbc log

(rankΠ c

C)+ 4h(εbc) (59)

Since xy le x minus y + 1 for x y isin [0 1]9 we find

2minus(12)I(AC|B)ρ minus 2minus(12)I(AC|B)ρbc

le 2minus(12)I(AC|B)ρ minus 2minus(12)tr(Π cCρC)I(AC|B)

ρbc minus tr(Π c

CρC)+ 1 (510)

According to (59) and since 2minusx ge 1 minus ln(2)x for x isin R we have

2minus(12)tr(Π cCρC)I(AC|B)

ρbc ge 2minus(12)I(AC|B)ρ2minus(12) tr(Π cCρC)

(8εbc log(rankΠ c

C)+4h(εbc))

ge 2minus(12)I(AC|B)ρ minus ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)

For two states σ1 and σ2 let P(σ1 σ2) =radic

1 minus F(σ1 σ2)2 denote the purified distance Applyingthe Fuchsndashvan de Graaf inequality [44] and a variant of the gentle-measurement lemma (seelemma 121 in the electronic supplementary material) gives

P(ρABC ρbc

ABC

)2 = 1 minus F(ρABC ρbc

ABC

)2 le 1 minus tr(Πb

B otimesΠ cCρBC

) (513)

Since the purified distance is a metric [45] that is monotonous under trace-preserving completelypositive maps [46 theorem 34] (513) gives

P (ρABCRBrarrBC(ρAB))le P(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)

9For x = 0 the statement clearly holds For (0 1] times [0 1] (x y) rarr f (x y) = xy minus x + y minus 1 isin R we find by using the convexityof y rarr f (x y) that maxxisin(01] maxyisin[01] f (x y) = 0


15


As the fidelity for states lies between zero and one (514) implies

F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2

le F (ρABCRBrarrBC(ρAB))2 + 4(

1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)

le F (ρABCRBrarrBC(ρAB))2 + 8radic

1 minus tr(Πb

B otimesΠ cCρBC

)

le(

F (ρABCRBrarrBC(ρAB))+ 2radic

2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that

F(ρbcABCRBrarrBC(ρbc

AB)) le F(ρABCRBrarrBC(ρAB)) + 2radic

2(1 minus tr(ΠbB otimesΠ c

CρBC))14 (516)

By definition of the quantity R(middot) (see equation (51)) the combination of (512) and (516)yields

R(ρbc) minus R(ρ)

le ln(2)2

(8εbc log(rankΠ cC) + 4h(εbc)) + (1 minus tr(Π c

CρC)) + 2radic


CρBC))14

= ξ bc (517)

where εbc is bounded by (58) By a variant of the gentle-measurement lemma (see lemma 122in the electronic supplementary material) we find limbrarrinfin tr(Πb

B otimesΠ cCρBC) = tr(Π c

CρC) for allc isin N and hence limbrarrinfin εbc = 0 for any c isin N Furthermore we have limcrarrinfin tr(Π c

CρC) = 1 andlimcrarrinfin limbrarrinfin tr(Πb

B otimesΠ cCρBC) = 1 which implies that limcrarrinfin limbrarrinfin ξ bc = 0 This proves the

assertion

By lemma 51 using the notation defined at the beginning of Step 1 we find

lim supcrarrinfin

lim supbrarrinfin

Rbc (S) = lim supcrarrinfin

lim supbrarrinfin

infρisinS

Rbc (ρ)

ge lim supcrarrinfin

lim supbrarrinfin

infρisinS

Rbc (ρbc) minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

inf

ρbcisinSbcRbc (ρbc)

minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)

where the second equality step is valid since all states in S have the same fixed marginal on B otimes Cand since the sequence ξ bcbcisinN only depends on this marginal The penultimate step uses thatlimcrarrinfin limbrarrinfin ξ bc = 0 The final inequality follows by definition of Rbc

BrarrBC Inequality (518)implies that there exist sequences bkkisinN and ckkisinN such that lim supkrarrinfinRbk ck (S) ge 0 SettingRk

BrarrBC =Rbk ckBrarrBC then implies that there exists a sequence Rk

BrarrBCkisinN of recovery maps thatsatisfies

lim supkrarrinfin

Rk (S) ge 0 (519)


16


(b) Step 2 Existence of a limitRecall that S is the set of density operators on A otimes B otimes C with a fixed marginal ρBC on B otimes C Thegoal of this step is to use (519) to prove that there exists a recovery map RBrarrBC such that

R(S) ge 0 (520)

Let ΠmB misinN and Πm

C misinN be sequences of projectors with rank m that weakly converge to idB

and idC respectively Furthermore for any m and any R isin TPCP(B B otimes C) let [R]m be the tracenon-increasing map obtained from R by projecting the input and output withΠm

B andΠmB otimesΠm

C respectively We start with a preparatory lemma that proves a relationship between [R]m (S) andR(S)

Lemma 52 For any ρBC isin D(B otimes C) there exists a sequence of reals δmmisinN with limmrarrinfin δm = 010

such that for any R isin TPCP(B B otimes C) we have

[R]m (S) ge R(S) minus δm minus 4ε14 (521)

where R(ρB) minus ρBC1 le ε

Proof For any ρABC isin S and any m isin N let us define the non-negative operator ρmAB = (idA otimes

ΠmB )ρAB (idA otimesΠm

B ) By definition of R(middot) (see equation (51)) it suffices to show that for anyρABC isin S any R isin TPCP(B B otimes C) ε isin [0 2] such that R(ρB) minus ρBC1 le ε and

ρmABC = (idA otimesΠm

B otimesΠmC )RBrarrBC(ρm

AB)(idA otimesΠmB otimesΠm

C ) (522)

we have F(ρABC ρmABC) ge F(ρABCRBrarrBC(ρAB)) minus δm minus 4ε14 As in Step 1 let P(middot middot) denote the

purified distance A variant of the gentle-measurement lemma (see lemma 121 in the electronicsupplementary material) implies that

P(ρAB ρmAB)2 = 1 minus F(ρAB ρm

AB)2 le 1 minus tr(ρBΠmB )2 (523)

Similarly we obtain

P(RBrarrBC(ρmAB) ρm

ABC)2 le 1 minus tr(RBrarrBC(ρmAB)Πm

B otimesΠmC )2

= 1 minus tr(RBrarrBC(ρmB )Πm

B otimesΠmC )2 (524)

By Houmllderrsquos inequality monotonicity of the trace norm for trace-preserving completely positivemaps [47 example 918 and corollary 9110] and (523) together with the Fuchsndashvan de Graafinequality [44] and a variant of the gentle-measurement lemma (see lemma 121 given in theelectronic supplementary material) we find

|tr((RBrarrBC(ρmB ) minus RBrarrBC(ρB))Πm

B otimesΠmC )| le RBrarrBC(ρB) minus RBrarrBC(ρm

B )1ΠmB otimesΠm

C infin

= RBrarrBC(ρB) minus RBrarrBC(ρmB )1 le ρB minus ρm

B 1

le ρAB minus ρmAB1 le 2

radic1 minus tr(ρBΠ

mB )2 (525)

Combining (524) (525) and Houmllderrsquos inequality together with the assumption R(ρB) minus ρBC1 leε gives


ABC)2 le 1 minus tr(RBrarrBC(ρB)ΠmB otimesΠm

C )2 + 4radic

1 minus tr(ρBΠmB )2

le 1 minus tr(ρBCΠmB otimesΠm

C )2 + 4radic

1 minus tr(ρBΠmB )2 + 2ε (526)

10The precise form of the sequence δmmisinN can be found in the proof (see equation (528))


17


Inequalities (523) (526) and the monotonicity of the purified distance under trace-preservingand completely positive maps [46 Theorem 34] show that

P(ρABC ρmABC) le P(ρABCRBrarrBC(ρAB)) + P(RBrarrBC(ρAB)RBrarrBC(ρm

AB)) + P(RBrarrBC(ρmAB) ρm

ABC)

le P(ρABCRBrarrBC(ρAB)) + P(ρAB ρmAB) + P(RBrarrBC(ρm

AB) ρmABC)

le P(ρABCRBrarrBC(ρAB)) + (δm)2

8+

radic2ε (527)

for

δm =radic

8

(radic1 minus tr(ρBΠ

mB )2 +

radic1 minus tr(ρBCΠ

mB otimesΠm

C )2 + 4radic


)12

(528)

As the purified distance between two states lies inside the interval [0 1] and since (δm)28 + radic2ε isin

[0 6] (527) implies that whenever F(ρABCRBrarrBC(ρAB))2 ge (δm)2 + 8radic

2ε we have

F(ρABC ρmABC)2 ge F(ρABCRBrarrBC(ρAB))2 minus (δm)2 minus 8

radic2ε

ge (F(ρABCRBrarrBC(ρAB)) minusradic

(δm)2 + 8radic

2ε)2 (529)

As a result we find

F(ρABC ρmABC) ge F(ρABCRBrarrBC(ρAB)) minus δm minus

radic8(2ε)14 (530)

which proves (521) sinceradic

8214 le 4Recall that B and C are separable Hilbert spaces and that Πm

B misinN and ΠmB otimesΠm

C misinN

converge weakly to idB and idB otimes idC respectively A variant of the gentle-measurement lemma(see lemma 122 given in the electronic supplementary material) thus shows that limmrarrinfin δm = 0since limmrarrinfin tr(ρBΠ

mB ) = 1 and limmrarrinfin tr(ρBCΠ

mB otimesΠm

C ) = 1

The following lemma proves that for sufficiently large m and a recovery map RBrarrBC that mapsρB to density operators that are close to ρBC the operator [R]m(ρAB) has a trace that is boundedfrom below by essentially one

Lemma 53 Let A B and C be separable Hilbert spaces For any density operator ρAB isin D(A otimes B) andany R isin TPCP(B B otimes C) we have

tr([R]m(ρAB)) ge tr(ΠmB otimesΠm

C ρBC) minus 2radic

1 minus tr(ΠmB ρB) minus R(ρB) minus ρBC1 (531)

Proof We first note that by Houmllderrsquos inequality and monotonicity of the trace norm for trace-preserving completely positive maps [47 example 918 and corollary 9110] we have

|tr(ΠmB otimesΠm

C (R(ρB) minus R(ΠmB ρBΠ

mB )))|

le R(ρB) minus R(ΠmB ρBΠ

mB )1 le ρB minusΠm

B ρBΠmB 1 (532)

Together with Houmllderrsquos inequality this implies

tr([R]m(ρAB)) = tr(ΠmB otimesΠm

C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))

ge tr(ΠmB otimesΠm

C R(ρB)) minus ρB minusΠmB ρBΠ

mB 1


C ρBC) minus ρB minusΠmB ρBΠ

mB 1 minus R(ρB) minus ρBC1 (533)


18


Combining a generalization of the Fuchsndashvan de Graaf inequality (see lemma 82 in the electronicsupplementary material) and a variant of the gentle-measurement lemma (see lemma 121 in theelectronic supplementary material) gives

ρB minusΠmB ρBΠ

mB 1 le 2

radic1 minus F(ρBΠm

B ρBΠmB )2 = 2

radicradicradicradic1 minus tr(ΠmB ρB)F

(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic

1 minus tr(ΠmB ρB) (534)

which together with (533) proves the assertion

According to (519) the mappings Rk satisfy

Rk (S) ge minusεk (535)

with εk ge 0 such that lim infkrarrinfin εk = 0 As explained in remark 23 by considering a stateρABC = ρA otimes ρBC isin S (535) implies F(ρBCRk(ρB)) ge minusεk + 1 Applying the Fuchsndashvan de Graafinequality [44] gives

ρBC minus Rk(ρB)1 le 2radicεk(2 minus εk) = εk (536)

where lim infkrarrinfin εk = 0 because lim infkrarrinfin εk = 0By lemma 52 we have

[Rk]m (S) ge Rk (S) minus 4(εk)14 minus δm (537)

Hence using our starting point (519)

lim supkrarrinfin

[Rk]m (S) ge lim supkrarrinfin

Rk (S) minus 4(εk)14 minus δm ge minusδm (538)

Because for any fixed m isin N the mappings [Rk]m for k isin N are all contained in the same finite-dimensional subspace (ie the set of trace non-increasing maps from operators on the supportof Πm

B to operators on the support of ΠmB otimesΠm

C ) and because the space of all such mappingsis compact (see remark 103 in the electronic supplementary material) for any fixed m isin N thereexists a subsequence of the sequence [Rk]mkisinN that converges Specifically for any fixed m isin N

there exists a sequence kmi iisinN such that

Rm = limirarrinfin

[Rkmi ]m (539)

is well defined Furthermore because of the continuity of R rarr R(ρABC) on the set of maps fromoperators on the support of Πm


C (see lemma 104 givenin the electronic supplementary material) we have

Rm (S) = infρisinS

Rm (ρ) = infρisinS

limirarrinfin

[Rkm

i ]m (ρ) ge lim supirarrinfin

infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm

i ]m (S) ge minusδm (540)

and hence

lim infmrarrinfin Rm (S) ge 0 (541)

Without loss of generality we can assume that the projector ΠmB is in the eigenbasis of ρB that

is denoted by |b〉mB For a basis |b〉m

B we define the projector Πm

B= WΠm

B Wdagger for an isometryW =sum

b |b〉〈b|mB

For any m isin N let ρmBCB

be the operator obtained by applying Rm to a purification

ρBB = (ρ12B otimes idB)

sumb |b〉B otimes |b〉B of ρB


19


As explained above there exists a converging subsequence km+1i iisinN of km+1

i iisinN such that

Rm = limirarrinfin[Rkm+1i ]m Using the definition of Rm and that Πm

B leΠmprimeB Πm

C leΠmprimeC and Πm

BleΠmprime

Bfor m le mprime we obtain

ρmBCB

= Rm(ρBB)

= limirarrinfin

[Rkm+1i ]m(ρBB) = lim

irarrinfin(Πm

B otimesΠmC )[Rkm+1

i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB

)[Rkm+1i ]m+1(ρBB)(Πm

B otimesΠmC otimesΠm

B)

= (ΠmB otimesΠm

C otimesΠmB

)Rm+1(ρBB)(ΠmB otimesΠm

C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)

As a result since ΠmB leΠmprime

B ΠmC leΠmprime

C and ΠmB

leΠmprimeB

for m le mprime we have for any m le mprime

ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)

A variant of the gentle-measurement lemma (see lemma 121 in the electronic supplementarymaterial) together with (543) implies

F(ρmBCB

ρmprimeBCB

) = F(ΠmB otimesΠm

C otimesΠmBρmprime

BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)

A generalization of the Fuchsndashvan de Graaf inequality (see lemma 82 in the electronicsupplementary material) yields for mprime ge m

ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)

We now prove that as m rarr infin tr(ρmBCB

) goes to 1 Note that since B is a separable Hilbert space and

ρBB is normalized it can be written as ρBB = |ψ〉〈ψ | where |ψ〉 is a state on B otimes B Furthermoreas Πm


Ble idBCB (543) implies that

tr(ρmBCB

) le tr(ρmprimeBCB

) le 1 for mprime ge m (546)

By definition of ρmBCB

lemma 53 together with (536) implies that

limmrarrinfin tr(ρm

BCB) = lim

mrarrinfin limirarrinfin

tr([Rkmi ]m(ρBB))

ge limmrarrinfin tr(Πm

B otimesΠmC ρBC) minus ρB minusΠm

B ρBΠmB 1 minus lim inf

irarrinfinεkm

i


B otimesΠmC ρBC) minus 2

radic1 minus tr(Πm

B ρB)2 = 1 (547)

where the second inequality uses a generalized version of the Fuchsndashvan de Graaf inequality(see lemma 82 in the electronic supplementary material) a variant of the gentle-measurementlemma (see lemma 121 in the electronic supplementary material) and that lim infirarrinfin εkm

i = 0for all m isin N The final step follows by another variant of the gentle-measurement lemma (seelemma 122 in the electronic supplementary material)

Equations (545)ndash(547) show that ρmBCB

misinN is a Cauchy sequence Because the set ofsub-normalized non-negative operators (ie the set of sub-normalized density operators)is complete11 this sequence converges towards such an operator ie we can define a

11We note that the set of sub-normalized density operators on a Hilbert space is clearly closed Since every Hilbert space iscomplete and as every closed subspace of a complete space is complete [42 ch II section 34 proposition 8] this implies thatthe set of sub-normalized density operators is complete


20


density operatorρBCB = lim

mrarrinfin ρmBCB

(548)

We note that the operators ρmBCB

are not normalized in general However (547) shows that ρBCBhas unit trace We now define the recovery map RBrarrBC as the one that maps ρBB to ρBCB Wenote that this does not uniquely define the recovery map RBrarrBC which is not a problem astheorem 21 proves the existence of a recovery map that satisfies (21) and does not claim thatthis map is unique It remains to show that RBrarrBC has the property (520) This follows from theobservation that any density operator ρAB can be obtained from the purification ρBB by applyinga trace-preserving completely positive map TBrarrA from B to A By a continuity property statedin lemma 105 in the electronic supplementary material and because TBrarrA commutes with anyrecovery map RBrarrBC from B to B otimes C we have

RBrarrBC(ρAB) = (RBrarrBC TBrarrA)(ρBB) = (TBrarrA RBrarrBC)(ρBB) = TBrarrA(ρBCB)

= TBrarrA( limmrarrinfin ρm

BCB) = lim

mrarrinfin TBrarrA(ρmBCB

) = limmrarrinfin(TBrarrA Rm

BrarrBC)(ρBB)

= limmrarrinfin(Rm

BrarrBC TBrarrA)(ρBB) = limmrarrinfin Rm

BrarrBC(ρAB) (549)

Using the continuity of the fidelity [1213] this implies that

R(ρ) = limmrarrinfin Rm (ρ) (550)

for any ρ isin S Combining this with (541) gives

R(S) = infρisinS

R(ρ) = infρisinS

limmrarrinfin Rm (ρ) ge lim inf

mrarrinfin infρisinS

Rm (ρ) = lim infmrarrinfin Rm (S) ge 0 (551)

which concludes Step 2 and thus completes the proof of theorem 21 in the general case where Band C are no longer finite-dimensional

6 Proof of corollary 24The first statement of corollary 24 that holds for separable Hilbert spaces follows immediatelyfrom theorem 21 since 2minus(12)I(AC|B)ρ ge 1 minus (ln(2)2)I(A C|B)ρ The proof of the second statementof corollary 24 is partitioned into three steps12 We first show that a similar method as used in sect4can be used to reveal certain insights about the structure of the recovery map RBrarrBC (which isnot universal) that satisfies

F(ρABCRBrarrBC(ρAB)) ge 1 minus ln(2)2

I(A C|B)ρ (61)

In a second step by invoking proposition 41 we use this knowledge to prove that for a fixedA system there exists a recovery map that satisfies (61) which is universal and preserves thestructure of the non-universal recovery map from before Finally in Step 3 we show how thedependency on the fixed A system can be removed

(a) Step 1 Structure of a non-universal recovery mapWe will show that for any density operator ρABC on A otimes B otimes C where A B and C are finite-dimensional Hilbert spaces there exists a trace-preserving completely positive map RBrarrBC thatsatisfies (61) and is of the form

XB rarr ρ12BC WBC(ρminus12

B XBρminus12B otimes idC)Wdagger

BCρ12BC (62)

on the support of ρB where WBC is a unitary on B otimes C We start by proving the followingpreparatory lemma

12Although corollary 24 does not immediately follow from theorem 21 it is justified to term it as such as it follows by thesame proof technique that is used to derive theorem 21 (in particular it makes use of proposition 41)


21


Lemma 61 For any density operator ρABC on A otimes B otimes C where A B and C are finite-dimensionalHilbert spaces there exists a trace-preserving completely positive map RBrarrBC of the form

XB rarr VBCρ12BC (ρminus12

B UBXBUdaggerBρ

minus12B otimes idC)ρ12

BC VdaggerBC (63)

where VBC is a unitary on B otimes C that commutes with ρBC and UB is a unitary on B that commutes withρB such that


I(A C|B)ρ (64)

Proof Let ρABC be an arbitrary state on A otimes B otimes C and let ρ0ABC be a Markov chain with the same

marginal on the B otimes C system ie ρ0BC = ρBC For p isin (0 1] define the state

ρp



The main result of [6] (see theorem 51 and remark 43 in [6]) implies that there exists a recoverymap RBrarrBC of the form


B UBXBUdaggerBρ


BC VdaggerBC (66)

where UB is diagonal with respect to the eigenbasis of ρB UBUdaggerB le idB and VBC is a unitary on

B otimes C such that

F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)

(Alternatively this statement also follows from [20]mdashwhich however appeared after thecompletion of this work) By lemma 62 using that I(A C|B)ρ0 = 0 since ρ0

ABC is a Markov chainthis may be rewritten as

p(1 minus F(ρABCRBrarrBC(ρAB))) + (1 minus p)(1 minus F(ρ0ABCRBrarrBC(ρ0

AB)))

le pln(2)

2I(A C|B)ρ (68)

Let us assume by contradiction that any recovery map RBrarrBC that satisfies (68) does not leaveρ0

ABC invariant ie ρ0ABC =RBrarrBC(ρ0

AB) This implies that there exists a δR isin (0 1] which maydepend on the recovery map RBrarrBC such that 1 minus F(ρ0

ABCRBrarrBC(ρ0AB)) = δR In the following

we argue that there exists a universal (ie independent of RBrarrBC) constant δ isin (0 1] such that1 minus F(ρ0

ABCRBrarrBC(ρ0AB)) ge δ for all recovery maps RBrarrBC that satisfy (68) Since the set of

trace-preserving completely positive maps from B to B otimes C that satisfy (68) is compact13 andthe function f TPCP(B B otimes C) RBrarrBC rarr 1 minus F(ρ0

ABCRBrarrBC(ρ0AB)) isin [0 1] is continuous (see

lemma 104 in the electronic supplementary material) Weierstrassrsquo theorem ensures that δ =minRBrarrBC f (RBrarrBC) where we optimize over the set of trace-preserving completely positive mapsfrom B to B otimes C that satisfy (68) exists By assumption for every recovery map RBrarrBC thatsatisfies (68) we have f (RBrarrBC)gt 0 and hence δ isin (0 1] If we insert any such recovery mapRBrarrBC into (68) this gives

1 minus F(ρABCRBrarrBC(ρAB)) + δ

pminus δ le ln(2)

2I(A C|B)ρ (69)

which cannot be valid for sufficiently small p To see this we note that (69) can be rewritten as

p ge δ

(ln(2)2)I(A C|B)ρ + δ + F(ρABCRBrarrBC(ρAB)) minus 1 (610)

since C is assumed to be a finite-dimensional system and as such I(A C|B)ρ ltinfin This contradictsour assumption that every recovery map that satisfies (68) does not leave ρ0

ABC invariant Sinceby [6] for any p isin (0 1] there exists a recovery map RBrarrBC of the form (63) that satisfies (68) we

13This set is bounded as the set of trace-preserving completely positive maps from B to B otimes C is bounded (see remark 103in the electronic supplementary material) Furthermore this set is closed since the set of trace-preserving completelypositive maps from B to B otimes C is closed (see remark 103 in the electronic supplementary material) and the mappingRBrarrBC rarr F(ρABCRBrarrBC(ρAB)) is continuous for all states ρABC (see lemma 104 in the electronic supplementary material)The HeinendashBorel theorem then implies compactness


22


conclude that there exists a recovery map RBrarrBC of the form (63) that satisfies (68) and leavesρ0

ABC invariant We note that for recovery maps that leave ρ0ABC invariant (68) simplifies to (64)

for all p Thus there exists a recovery map RBrarrBC of the form (63) satisfying (64) that leavesρ0

ABC invariant ie RBrarrBC(ρ0AB) = ρ0

ABC Since ρ0ABC = ρA otimes ρBC is a Markov chain with marginal

ρ0BC = ρBC the condition RBrarrBC(ρ0

AB) = ρ0ABC implies that RBrarrBC(ρB) = ρBC

We have thus shown that there exists a recovery map RBrarrBC that satisfies (64) and fulfils

RBrarrBC(ρB) = VBCρ12BC (UBUdagger

B otimes idC)ρ12BC Vdagger

BC = ρBC (611)

Using the fact that RBrarrBC is trace preserving shows that

idB = trC(UdaggerBρ

minus12B ρ

12BC VBCVdagger

BCρ12BC ρ

minus12B UB) = Udagger

Bρminus12B ρBρ

minus12B UB = Udagger

BUB (612)

This simplifies (611) to VBCρBCVdaggerBC = ρBC ie VBC and ρBC commute which concludes the proof

Lemma 61 implies that the mapping (63) can be written as



BCρ12BC (613)

with WBC = VBCUB otimes idC which is a unitary as VBC and UB are unitaries Furthermore WBC issuch that (613) is trace-preserving

(b) Step 2 Structure of a universal recovery map for fixed A systemIn this step we show that the recovery map satisfying (61) of the form (62) whose existencehas been established in Step 1 can be made universal without sacrificing the (partial) knowledgeabout its structure The idea is to apply proposition 41 for the function family

R(ρ) D(A otimes B otimes C) rarr R cup minusinfin

ρABC rarr F(ρABCRBrarrBC(ρAB)) minus 1 + ln(2)2

I(A C|B)ρ (614)

We therefore need to verify that the assumptions of proposition 41 are fulfilled This is done bythe following lemma We first note that since C is finite-dimensional this implies that R(ρ)ltinfinfor all ρ isin D(A otimes B otimes C)


Proof We start by showing that R(middot) satisfies property (i) For ρp

AABCas defined in (44)

we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)

)+ pF (ρABCRBrarrBC(ρAB)) (615)



RBrarrBC(ρp



The relevant density operators thus satisfy the orthogonality conditions for equality in lemma 81given in the electronic supplementary material from which (615) follows Furthermore asexplained in the proof of lemma 42 we have



R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23


We next verify that R(middot) fulfils property (ii) Let RBrarrBCRprimeBrarrBC isin TPCP(B B otimes C) α isin [0 1]

and RBrarrBC = αRBrarrBC + (1 minus α)RprimeBrarrBC A specific property of the fidelity stated in lemma 81 in

the electronic supplementary material implies that for any state ρABC on A otimes B otimes C

F(ρABC RBrarrBC(ρAB)) = F(ρABCαRBrarrBC(ρAB) + (1 minus α)RprimeBrarrBC(ρAB))

ge αF(ρABCRBrarrBC(ρAB)) + (1 minus α)F(ρABCRprimeBrarrBC(ρAB)) (619)

and hence by the definition of R(middot)R(ρ) ge αR(ρ) + (1 minus α)Rprime (ρ) (620)

The function ρ rarr R(ρ) is continuous which clearly implies property (iii) To see thisrecall that by the AlickindashFannes inequality ρ rarr I(A C|B)ρ is continuous for a finite-dimensionalC system [35] Furthermore since ρAB rarrRBC(ρAB) is continuous (see lemma 105 in theelectronic supplementary material) the continuity of the fidelity [1213] implies that ρABC rarrF(ρABCRBrarrBC(ρAB)) is continuous which then establishes property (iii)

Finally it remains to show that R(middot) satisfies property (iv) which however follows directlyby lemma 104 given in the electronic supplementary material

Let P sube TPCP(B B otimes C) be the convex hull of the set of trace-preserving completely positivemappings from the B to the B otimes C system that are of the form (62) We note that the elements ofP are mappings of the form (24) since a convex combination of unitary mappings are unital anda convex combination of trace-preserving maps remains trace-preserving Proposition 41 whichis applicable as shown in lemma 62 together with Step 1 therefore proves the assertion for a fixedfinite-dimensional A system

(c) Step 3 Independence from the A systemLet S be the set of all density operators on A otimes B otimes C with a fixed marginal ρBC on B otimes C whereB and C are finite-dimensional Hilbert spaces and A is the infinite-dimensional Hilbert space 2

of square summable sequencesWe note that the set of trace-preserving completely positive maps of the form (24) on finite-

dimensional systems is compact which follows by remark 103 (see the electronic supplementarymaterial) together with the fact that the intersection of a compact set and a closed set is compactHence using lemma 62 (in particular properties (iii) and (iv)) and the result from Step 2 the sameargument as in Step 4 of sect4 can be applied to conclude the existence of a recovery map RBrarrBC ofthe form (24) such that R(S) ge 0

As every separable Hilbert space A can isometrically embedded into A [21 Theorem II7] andsince R is invariant under isometries applied on the extension space A we can conclude thatthe recovery map RBrarrBC remains valid for any separable extension space A This proves thestatement of corollary 24 for finite-dimensional B and C systems

7 DiscussionOur main result is that for any density operator ρBC on B otimes C there exists a recovery map RBrarrBC

such that the distance between any extension ρABC of ρBC acting on A otimes B otimes C and RBrarrBC(ρAB)is bounded from above by the conditional mutual information I(A C|B)ρ It is natural to askwhether such a map can be described as a simple and explicit function of ρBC In fact it wasconjectured in [511] that (12) holds for a very simple choice of map namely

TBrarrBC XB rarr ρ12BC (ρminus12

B XBρminus12B otimes idC)ρ12

BC (71)

called the transpose map or Petz recovery map This conjecture if correct would have importantconsequences in obtaining remainder terms for the monotonicity of the relative entropy [19] Asdiscussed in the Introduction if ρABC is such that it is a (perfect) quantum Markov chain or the Bsystem is classical the claim of the conjecture is known to hold


24


One possible approach to prove a result of this form would be to start from the result (12) foran unknown recovery map and then show that the transpose map TBrarrBC cannot be much worsethan any other recovery map In fact a theorem of Barnum amp Knill [48] directly implies that whenρABC is pure we have

F(ρABC TBrarrBC(ρAB)) le F(A C|B)ρ leradic

F(ρABC TBrarrBC(ρAB)) (72)

This shows that if ρABC is pure an inequality of the form (12) with the fidelity replaced byits square root holds for the transpose map In order to generalize this to all states one mighthope that (72) also holds for mixed states ρABC However this turns out to be wrong even whenthe state ρABC is completely classical (see sect13 in the electronic supplementary material for anexample)

Another interesting question is whether the lower bound in terms of the measured relativeentropy (22) can be improved to a relative entropy Such an inequality is known to be false ifwe restrict the recovery map to be the transpose map (71) [9] but it might be true when weoptimize over all recovery maps It is worth noting that in case such an inequality holds for anyρABC and a corresponding recovery map then the argument presented in this work would implythat there exists a universal recovery map satisfying (22) with the relative entropy instead of themeasured relative entropy This can be seen by defining the function family ρ rarrR(ρ) = I(A C|B)ρ minus D(ρABCRBrarrBC(ρAB)) A linearity property of the relative entropy for orthogonal states(see lemma 92 in the electronic supplementary material) the convexity of the relative entropy[49 theorem 1112] and the lower semicontinuity of the relative entropy [41 example 722] implythat R(middot) satisfies properties (i)ndash(iv) As a result proposition 41 is applicable which can be usedto prove the existence of a universal recovery map

After the completion of this work there was a series of works around finding improvements oralternative proofs for inequality (12) In [20] an alternative proof for (12) based on the Hadamardthree-line theorem was discovered14 After that yet another proof for (12) has been found which isbased elementary properties of pinching maps and the operator logarithm [50] Finally in [40] anexplicit and universal recovery map has been determined that satisfies (12) based on Hirschmanrsquosstrengthening [51] of the Hadamard three-line theorem

Data accessibility This work does not have any experimental dataAuthorsrsquo contributions All authors contributed equally to this workCompeting interests We have no competing interestsFunding This project was supported by the European Research Council (ERC) via grant no 258932 by theSwiss National Science Foundation (SNSF) via the National Centre of Competence in Research lsquoQSITrsquo and bythe European Commission via the project lsquoRAQUELrsquoAcknowledgements We thank Mario Berta Fernando Brandatildeo Philipp Kammerlander Joseph Renes VolkherScholz Marco Tomamichel and Mark Wilde for discussions about approximate Markov chains

References1 Petz D 1986 Sufficient subalgebras and the relative entropy of states of a von Neumann

algebra Commun Math Phys 105 123ndash131 (doi101007BF01212345)2 Petz D 1986 Sufficiency of channels over von Neumann algebras Q J Math Oxf 39 97ndash108

(doi101093qmath39197)3 Hayden P Jozsa R Petz D Winter A 2004 Structure of states which satisfy strong

subadditivity of quantum entropy with equality Commun Math Phys 246 359ndash374 (doi101007s00220-004-1049-z)

4 Datta N Wilde MM 2015 Quantum Markov chains sufficiency of quantum channels andReacutenyi information measures J Phys A Math Theor 48 505301 (doi1010881751-81134850505301)

14The result in [20] is more general than (12) as it proves a remainder term for the monotonicity of the relative entropy whichimplies (12) as a special case (see [20] for more details)


25


5 Kim IH 2013 Application of conditional independence to gapped quantum many-bodysystems See httpwwwphysicsusydeduauquantumCoogee2013PresentationsKimpdf

6 Fawzi O Renner R 2015 Quantum conditional mutual information and approximate Markovchains Commun Math Phys 340 575ndash611 (doi101007s00220-015-2466-x)

7 Christandl M Schuch N Winter A 2012 Entanglement of the antisymmetric state CommunMath Phys 311 397ndash422 (doi101007s00220-012-1446-7)

8 Ibinson B Linden N Winter A 2008 Robustness of quantum Markov chains Commun MathPhys 277 289ndash304 (doi101007s00220-007-0362-8)

9 Winter A Li K 2012 A stronger subadditivity relation With applications to squashedentanglement sharability and separability See httpwwwmathsbrisacukcsajwstronger_subadditivitypdf

10 Zhang L 2013 Conditional mutual information and commutator Int J Theor Phys 522112ndash2117 (doi101007s10773-013-1505-7)

11 Berta M Seshadreesan KP Wilde MM 2015 Reacutenyi generalizations of the conditional quantummutual information J Math Phys 56 022205 (doi10106314908102)

12 Uhlmann A 1976 The lsquotransition probabilityrsquo in the state space of a lowast-algebra Rep Math Phys9 273ndash279 (doi1010160034-4877(76)90060-4)

13 Jozsa R 1994 Fidelity for mixed quantum states J Mod Opt 41 2315ndash2323 (doi10108009500349414552171)

14 Seshadreesan KP Wilde MM 2015 Fidelity of recovery squashed entanglement andmeasurement recoverability Phys Rev A 92 042321 (doi101103PhysRevA92042321)

15 Berta M Tomamichel M 2015 The fidelity of recovery is multiplicative (httparxivhttparxivorgabs150207973)

16 Brandatildeo FGSL Harrow AW Oppenheim J Strelchuk S 2015 Quantum conditional mutualinformation reconstructed states and state redistribution Phys Rev Lett 115 050501(doi101103PhysRevLett115050501)

17 Hiai F Petz D 1991 The proper formula for relative entropy and its asymptotics in quantumprobability Commun Math Phys 143 99ndash114 (doi101007BF02100287)

18 Hayashi M 2001 Asymptotics of quantum relative entropy from a representation theoreticalviewpoint J Phys A Math Gen 34 3413 (doi1010880305-44703416309)

19 Berta M Lemm M Wilde M 2015 Monotonicity of quantum relative entropy andrecoverability Quant Inform Comput 15 1333ndash1354

20 Wilde MM 2015 Recoverability in quantum information theory Proc R Soc A 471 20150338(doi101098rspa20150338)

21 Reed M Simon B 1980 Functional analysis Amsterdam The Netherlands Elsevier22 Einsiedler M Ward T 2010 Ergodic theory Berlin Germany Springer23 Lieb E Ruskai M 1973 Proof of the strong subadditivity of quantum-mechanical entropy

J Math Phys 14 1938ndash1941 (doi10106311666274)24 Lieb EH Ruskai MB 1973 A fundamental property of quantum-mechanical entropy Phys

Rev Lett 30 434ndash436 (doi101103PhysRevLett30434)25 Li K Winter A 2014 Squashed entanglement k-extendibility quantum Markov chains and

recovery maps (httparxivorgabs14104184)26 Kim IH 2013 Conditional independence in quantum many-body systems PhD thesis

Caltech27 Brandatildeo FG Harrow AW 2013 Product-state approximations to quantum ground states In

Proc of the 45th Annual ACM Symp on Theory of Computing STOC rsquo13 Palo Alto CA 1ndash4 Junepp 871ndash880 New York NY ACM

28 Brandatildeo FG Harrow AW 2013 Quantum de Finetti theorems under local measurements withapplications In Proc of the 45th Annual ACM Symposium on Theory of Computing STOC rsquo13Palo Alto CA 1ndash4 June pp 861ndash870 New York NY ACM

29 Jain R Radhakrishnan J Sen P 2003 A lower bound for the bounded round quantumcommunication complexity of set disjointness In IEEE 44th Annual IEEE Symp on Foundationsof Computer Science Cambridge MA 11ndash14 October pp 220ndash229 Piscataway NJ IEEE

30 Kerenidis I Laplante S Lerays V Roland J Xiao D 2012 Lower bounds on informationcomplexity via zero-communication protocols and applications In IEEE 53rd Annual Sympon Foundations of Computer Science New Brunswick NJ 20ndash23 October pp 500ndash509 PiscatawayNJ IEEE


26


31 Touchette D 2014 Quantum information complexity and amortized communication(httparxivorgabs14043733)

32 Bravyi S Hastings MB Verstraete F 2006 Lieb-Robinson bounds and the generationof correlations and topological quantum order Phys Rev Lett 97 050401 (doi101103PhysRevLett97050401)

33 Kitaev A Preskill J 2006 Topological entanglement entropy Phys Rev Lett 96 110404(doi101103PhysRevLett96110404)

34 Levin M Wen X-G 2006 Detecting topological order in a ground state wave function PhysRev Lett 96 110405 (doi101103PhysRevLett96110405)

35 Alicki R Fannes M 2004 Continuity of quantum conditional information J Phys A Math Gen37 55ndash57 (doi1010880305-4470375L01)

36 Muumlller-Lennert M Dupuis F Szehr O Fehr S Tomamichel M 2013 On quantumReacutenyi entropies a new generalization and some properties J Math Phys 54 122203(doi10106314838856)

37 Wilde MM Winter A Yang D 2014 Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Reacutenyi relative entropy Commun MathPhys 331 593ndash622 (doi101007s00220-014-2122-x)

38 Fuchs CA 1996 Distinguishability and accessible information in quantum theory PhD thesisUniversity of New Mexico (httparxivorgabsquant-ph9601020)

39 Sion M 1958 On general minimax theorems Pacific J Math 8 171 (doi102140pjm19588171)

40 Junge M Renner R Sutter D Wilde MM Winter A 2015 Universal recovery from a decreaseof quantum relative entropy (httparxivorgabs150907127)

41 Holevo AS 2012 Quantum systems channels information De Gruyter Studies in MathematicalPhysics 16 Berlin Germany De Gruyter

42 Bourbaki N 1966 Elements of mathematics general topology Hermann MO Eacutediteures desSciences et des Arts

43 Furrer F Aringberg J Renner R 2011 Min- and max-entropy in infinite dimensions CommunMath Phys 306 165ndash186 (doi101007s00220-011-1282-1)

44 Fuchs CA van de Graaf J 1999 Cryptographic distinguishability measures for quantum-mechanical states IEEE Trans Inform Theory 45 1216ndash1227 (doi10110918761271)

45 Tomamichel M Colbeck R Renner R 2010 Duality between smooth min- and max-entropiesIEEE Trans Inform Theory 56 4674ndash4681 (doi101109TIT20102054130)

46 Tomamichel M 2012 A framework for non-asymptotic quantum information theory PhDthesis ETH Zurich (httparxivorgabs12032142)

47 Wilde M 2013 Quantum information theory Cambridge UK Cambridge University Press48 Barnum H Knill E 2002 Reversing quantum dynamics with near-optimal quantum and

classical fidelity J Math Phys 43 2097ndash2106 (doi10106311459754)49 Nielsen MA Chuang IL 2000 Quantum computation and quantum information Cambridge UK

Cambridge University Press50 Sutter D Tomamichel M Harrow AW 2015 Monotonicity of relative entropy via pinched Petz

recovery map (httparxivorgabs150700303)51 Hirschman II 1952 A convexity theorem for certain groups of transformations J drsquoAnalyse

Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications

Proof for finite dimensions

Step 1 Proof of proposition 41 for finite size sets S

Step 2 Extension to infinite sets S

Step 3 From proposition 41 to theorem 21 for fixed system A

Step 4 Independence from the A system

Extension to infinite dimensions

Step 1 Existence of a sequence of recovery maps

Step 2 Existence of a limit

Proof of corollary 24

Step 1 Structure of a non-universal recovery map

Step 2 Structure of a universal recovery map for fixed A system


Discussion

References

2


Very recently it has been shown that Markov states can be alternatively characterized byhaving a generalized Reacutenyi conditional mutual information that vanishes [4]

A natural question that is relevant for applications is whether the above statements are robust(In [5] an example is discussed that illustrates why this question is relevant In [6] furtherexplanations are given to emphasize the importance of this problem) Specifically one wouldlike to have a characterization of the states that have a small (but not necessarily vanishing)conditional mutual information ie I(A C|B) le ε for ε gt 0 First results revealed that such statescan have a large distance to Markov chains that is independent of ε [78] which has been takenas an indication that their characterization may be difficult However it has subsequently beenrealized that a more appropriate measure instead of the distance to a (perfect) Markov chain is toconsider how well (11) is satisfied [59ndash11] This motivated the definition of approximate Markovchains as states where (11) approximately holds

In recent work [6] it has been shown that the set of approximate Markov chains indeedcoincides with the set of states with small conditional mutual information In particular thedistance between the two terms in (11) which may be measured in terms of their fidelity F isbounded by the conditional mutual information1 More precisely for any state ρABC there exists atrace-preserving completely positive map RBrarrBC (the recovery map) such that


Furthermore a converse inequality of the form I(A C|B)2ρ le minusc2 log2 F(ρABCRBrarrBC(ρAB)) where

c depends logarithmically on the dimension of A can be shown to hold [611]We also note that the fidelity term in (12) maximized over all recovery maps ie

F(A C|B)ρ = supRBrarrBC

F(ρABCRBrarrBC(ρAB)) (13)

is called fidelity of recovery2 and has been introduced and studied in [1415] With this quantity themain result of [6] can be written as

I(A C|B)ρ ge minus2 log2 F(A C|B)ρ (14)

The fidelity of recovery has several natural properties eg it is monotonous under localoperations on A and C and it is multiplicative [15]

The result of [6] has been extended in various ways Based on quantum state redistributionprotocols it has been shown in [16] that (12) still holds if the fidelity term is replaced by themeasured relative entropy DM(middot middot) which is generally larger ie there exists a recovery map RBrarrBC

such that

I(A C|B)ρ ge DM(ρABCRBrarrBC(ρAB)) ge minus2 log2 F(ρABCRBrarrBC(ρAB)) (15)

The measured relative entropy is defined as the supremum of the relative entropy with measuredinputs over all projective measurements3 M= Mx ie

DM(ρσ ) = sup

D(M(ρ)M(σ )) M(ρ) =

sumx

tr(ρMx)|x〉〈x| withsum

xMx = id

(16)

where |x〉 is a finite set of orthonormal vectors This quantity was studied in [1718]Furthermore in [15] an alternative proof of (12) has been derived that uses properties of the

fidelity of recovery (in particular multiplicativity) Another recent work [19] showed how togeneralize ideas from [6] to prove a remainder term for the monotonicity of the relative entropyin terms of a recovery map that satisfies (12)

1The fidelity of ρ and σ is defined as F(ρ σ ) = radicρradicσ1

2We note that if A B and C are finite-dimensional Hilbert spaces the supremum is achieved since the set of recovery maps iscompact (see remark 103 in the electronic supplementary material) and the fidelity is continuous in the input state (see [1213]or lemma B9 in [6])3Without loss of generality these can be assumed to be rank-one projectors


3




XB rarr VBCρ12BC

(ρ



)ρ

12BC Vdagger

BC (17)
















)(23)


4




12BC UBCrarrBC

(ρ

minus12B XBρ

minus12B otimes idC

)ρ

12BC (24)



XB rarrint

VsBCρ

12BC(ρ

minus12B Us

BXBUsdaggerB ρ

minus12B otimes idC

)ρ

12BC Vsdagger

BCμ(ds) (25)






B andρit




) (26)




I(A C|B)ρ ge 0 (31)





5


A C

B

B





i =trGρ





AB)) (32)








6





D12(M(ρ)M(σ ))










R(S) = infρisinS

R(ρ) (43)



ρp





Sp =ρ

p

AABC ρABC isin S

(45)








7




R(Sp) = infρisinS


R(ρ) = (1 minus p)R(ρ0) + pR(S) (46)


R(S) = infρisinS

R(ρ) ge infρisinS


ge α infρisinS















Rp (Sp) ge 0 (410)





8


This implies that

Rp (ρ0) ge 0 or Rp (S) ge 0 (412)


Ru (ρ0) ge 0 and Rv (S) ge 0 (413)










R(ρ)ltinfin (416)







lim supnrarrinfin





lim supnrarrinfin


minρisinScupρ0

Rδn (ρ)

= lim supnrarrinfin



Λ0 =Ru (ρ0) and Λ1 =Rv (S) (421)

as well as

Λ0 =Rv (ρ0) and Λ1 =Ru (S) (422)


(1 minus u)Λ0 + uΛ1 ge 0 (423)

Similarly we have

(1 minus v)Λ0 + vΛ1 ge 0 (424)


9




1 minus vΛ1 (425)



Λ0 + (1 minus α)Λ1 (426)





and

minus α1 minus u






minus 1 minus v

v

∣∣∣∣le 4δ (430)


minus α1 minus u


(1 minus v

v+ 4δ







1 minus vminus u

1 minus u

∣∣∣∣lt 4δ (433)




(u

1 minus u+ 4δ






10








lim supmrarrinfin


infρisinSεn

Rεm (ρ)


infρisinSεm


Rεm (Sεm ) ge 0 (436)


R(ρ) =R(

limnrarrinfin ρn

)ge lim sup













As a result we find



11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

3




XB rarr VBCρ12BC

(ρ



)ρ

12BC Vdagger

BC (17)
















)(23)


4




12BC UBCrarrBC

(ρ

minus12B XBρ

minus12B otimes idC

)ρ

12BC (24)



XB rarrint

VsBCρ

12BC(ρ

minus12B Us

BXBUsdaggerB ρ

minus12B otimes idC

)ρ

12BC Vsdagger

BCμ(ds) (25)






B andρit




) (26)




I(A C|B)ρ ge 0 (31)





5


A C

B

B





i =trGρ





AB)) (32)








6





D12(M(ρ)M(σ ))










R(S) = infρisinS

R(ρ) (43)



ρp





Sp =ρ

p

AABC ρABC isin S

(45)








7




R(Sp) = infρisinS


R(ρ) = (1 minus p)R(ρ0) + pR(S) (46)


R(S) = infρisinS

R(ρ) ge infρisinS


ge α infρisinS















Rp (Sp) ge 0 (410)





8


This implies that

Rp (ρ0) ge 0 or Rp (S) ge 0 (412)


Ru (ρ0) ge 0 and Rv (S) ge 0 (413)










R(ρ)ltinfin (416)







lim supnrarrinfin





lim supnrarrinfin


minρisinScupρ0

Rδn (ρ)

= lim supnrarrinfin



Λ0 =Ru (ρ0) and Λ1 =Rv (S) (421)

as well as

Λ0 =Rv (ρ0) and Λ1 =Ru (S) (422)


(1 minus u)Λ0 + uΛ1 ge 0 (423)

Similarly we have

(1 minus v)Λ0 + vΛ1 ge 0 (424)


9




1 minus vΛ1 (425)



Λ0 + (1 minus α)Λ1 (426)





and

minus α1 minus u






minus 1 minus v

v

∣∣∣∣le 4δ (430)


minus α1 minus u


(1 minus v

v+ 4δ







1 minus vminus u

1 minus u

∣∣∣∣lt 4δ (433)




(u

1 minus u+ 4δ






10








lim supmrarrinfin


infρisinSεn

Rεm (ρ)


infρisinSεm


Rεm (Sεm ) ge 0 (436)


R(ρ) =R(

limnrarrinfin ρn

)ge lim sup













As a result we find



11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

4




12BC UBCrarrBC

(ρ

minus12B XBρ

minus12B otimes idC

)ρ

12BC (24)



XB rarrint

VsBCρ

12BC(ρ

minus12B Us

BXBUsdaggerB ρ

minus12B otimes idC

)ρ

12BC Vsdagger

BCμ(ds) (25)






B andρit




) (26)




I(A C|B)ρ ge 0 (31)





5


A C

B

B





i =trGρ





AB)) (32)








6





D12(M(ρ)M(σ ))










R(S) = infρisinS

R(ρ) (43)



ρp





Sp =ρ

p

AABC ρABC isin S

(45)








7




R(Sp) = infρisinS


R(ρ) = (1 minus p)R(ρ0) + pR(S) (46)


R(S) = infρisinS

R(ρ) ge infρisinS


ge α infρisinS















Rp (Sp) ge 0 (410)





8


This implies that

Rp (ρ0) ge 0 or Rp (S) ge 0 (412)


Ru (ρ0) ge 0 and Rv (S) ge 0 (413)










R(ρ)ltinfin (416)







lim supnrarrinfin





lim supnrarrinfin


minρisinScupρ0

Rδn (ρ)

= lim supnrarrinfin



Λ0 =Ru (ρ0) and Λ1 =Rv (S) (421)

as well as

Λ0 =Rv (ρ0) and Λ1 =Ru (S) (422)


(1 minus u)Λ0 + uΛ1 ge 0 (423)

Similarly we have

(1 minus v)Λ0 + vΛ1 ge 0 (424)


9




1 minus vΛ1 (425)



Λ0 + (1 minus α)Λ1 (426)





and

minus α1 minus u






minus 1 minus v

v

∣∣∣∣le 4δ (430)


minus α1 minus u


(1 minus v

v+ 4δ







1 minus vminus u

1 minus u

∣∣∣∣lt 4δ (433)




(u

1 minus u+ 4δ






10








lim supmrarrinfin


infρisinSεn

Rεm (ρ)


infρisinSεm


Rεm (Sεm ) ge 0 (436)


R(ρ) =R(

limnrarrinfin ρn

)ge lim sup













As a result we find



11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

5


A C

B

B





i =trGρ





AB)) (32)








6





D12(M(ρ)M(σ ))










R(S) = infρisinS

R(ρ) (43)



ρp





Sp =ρ

p

AABC ρABC isin S

(45)








7




R(Sp) = infρisinS


R(ρ) = (1 minus p)R(ρ0) + pR(S) (46)


R(S) = infρisinS

R(ρ) ge infρisinS


ge α infρisinS















Rp (Sp) ge 0 (410)





8


This implies that

Rp (ρ0) ge 0 or Rp (S) ge 0 (412)


Ru (ρ0) ge 0 and Rv (S) ge 0 (413)










R(ρ)ltinfin (416)







lim supnrarrinfin





lim supnrarrinfin


minρisinScupρ0

Rδn (ρ)

= lim supnrarrinfin



Λ0 =Ru (ρ0) and Λ1 =Rv (S) (421)

as well as

Λ0 =Rv (ρ0) and Λ1 =Ru (S) (422)


(1 minus u)Λ0 + uΛ1 ge 0 (423)

Similarly we have

(1 minus v)Λ0 + vΛ1 ge 0 (424)


9




1 minus vΛ1 (425)



Λ0 + (1 minus α)Λ1 (426)





and

minus α1 minus u






minus 1 minus v

v

∣∣∣∣le 4δ (430)


minus α1 minus u


(1 minus v

v+ 4δ







1 minus vminus u

1 minus u

∣∣∣∣lt 4δ (433)




(u

1 minus u+ 4δ






10








lim supmrarrinfin


infρisinSεn

Rεm (ρ)


infρisinSεm


Rεm (Sεm ) ge 0 (436)


R(ρ) =R(

limnrarrinfin ρn

)ge lim sup













As a result we find



11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

6





D12(M(ρ)M(σ ))










R(S) = infρisinS

R(ρ) (43)



ρp





Sp =ρ

p

AABC ρABC isin S

(45)








7




R(Sp) = infρisinS


R(ρ) = (1 minus p)R(ρ0) + pR(S) (46)


R(S) = infρisinS

R(ρ) ge infρisinS


ge α infρisinS















Rp (Sp) ge 0 (410)





8


This implies that

Rp (ρ0) ge 0 or Rp (S) ge 0 (412)


Ru (ρ0) ge 0 and Rv (S) ge 0 (413)










R(ρ)ltinfin (416)







lim supnrarrinfin





lim supnrarrinfin


minρisinScupρ0

Rδn (ρ)

= lim supnrarrinfin



Λ0 =Ru (ρ0) and Λ1 =Rv (S) (421)

as well as

Λ0 =Rv (ρ0) and Λ1 =Ru (S) (422)


(1 minus u)Λ0 + uΛ1 ge 0 (423)

Similarly we have

(1 minus v)Λ0 + vΛ1 ge 0 (424)


9




1 minus vΛ1 (425)



Λ0 + (1 minus α)Λ1 (426)





and

minus α1 minus u






minus 1 minus v

v

∣∣∣∣le 4δ (430)


minus α1 minus u


(1 minus v

v+ 4δ







1 minus vminus u

1 minus u

∣∣∣∣lt 4δ (433)




(u

1 minus u+ 4δ






10








lim supmrarrinfin


infρisinSεn

Rεm (ρ)


infρisinSεm


Rεm (Sεm ) ge 0 (436)


R(ρ) =R(

limnrarrinfin ρn

)ge lim sup













As a result we find



11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

7




R(Sp) = infρisinS


R(ρ) = (1 minus p)R(ρ0) + pR(S) (46)


R(S) = infρisinS

R(ρ) ge infρisinS


ge α infρisinS















Rp (Sp) ge 0 (410)





8


This implies that

Rp (ρ0) ge 0 or Rp (S) ge 0 (412)


Ru (ρ0) ge 0 and Rv (S) ge 0 (413)










R(ρ)ltinfin (416)







lim supnrarrinfin





lim supnrarrinfin


minρisinScupρ0

Rδn (ρ)

= lim supnrarrinfin



Λ0 =Ru (ρ0) and Λ1 =Rv (S) (421)

as well as

Λ0 =Rv (ρ0) and Λ1 =Ru (S) (422)


(1 minus u)Λ0 + uΛ1 ge 0 (423)

Similarly we have

(1 minus v)Λ0 + vΛ1 ge 0 (424)


9




1 minus vΛ1 (425)



Λ0 + (1 minus α)Λ1 (426)





and

minus α1 minus u






minus 1 minus v

v

∣∣∣∣le 4δ (430)


minus α1 minus u


(1 minus v

v+ 4δ







1 minus vminus u

1 minus u

∣∣∣∣lt 4δ (433)




(u

1 minus u+ 4δ






10








lim supmrarrinfin


infρisinSεn

Rεm (ρ)


infρisinSεm


Rεm (Sεm ) ge 0 (436)


R(ρ) =R(

limnrarrinfin ρn

)ge lim sup













As a result we find



11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

8


This implies that

Rp (ρ0) ge 0 or Rp (S) ge 0 (412)


Ru (ρ0) ge 0 and Rv (S) ge 0 (413)










R(ρ)ltinfin (416)







lim supnrarrinfin





lim supnrarrinfin


minρisinScupρ0

Rδn (ρ)

= lim supnrarrinfin



Λ0 =Ru (ρ0) and Λ1 =Rv (S) (421)

as well as

Λ0 =Rv (ρ0) and Λ1 =Ru (S) (422)


(1 minus u)Λ0 + uΛ1 ge 0 (423)

Similarly we have

(1 minus v)Λ0 + vΛ1 ge 0 (424)


9




1 minus vΛ1 (425)



Λ0 + (1 minus α)Λ1 (426)





and

minus α1 minus u






minus 1 minus v

v

∣∣∣∣le 4δ (430)


minus α1 minus u


(1 minus v

v+ 4δ







1 minus vminus u

1 minus u

∣∣∣∣lt 4δ (433)




(u

1 minus u+ 4δ






10








lim supmrarrinfin


infρisinSεn

Rεm (ρ)


infρisinSεm


Rεm (Sεm ) ge 0 (436)


R(ρ) =R(

limnrarrinfin ρn

)ge lim sup













As a result we find



11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

9




1 minus vΛ1 (425)



Λ0 + (1 minus α)Λ1 (426)





and

minus α1 minus u






minus 1 minus v

v

∣∣∣∣le 4δ (430)


minus α1 minus u


(1 minus v

v+ 4δ







1 minus vminus u

1 minus u

∣∣∣∣lt 4δ (433)




(u

1 minus u+ 4δ






10








lim supmrarrinfin


infρisinSεn

Rεm (ρ)


infρisinSεm


Rεm (Sεm ) ge 0 (436)


R(ρ) =R(

limnrarrinfin ρn

)ge lim sup













As a result we find



11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

10








lim supmrarrinfin


infρisinSεn

Rεm (ρ)


infρisinSεm


Rεm (Sεm ) ge 0 (436)


R(ρ) =R(

limnrarrinfin ρn

)ge lim sup













As a result we find



11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

11




RBrarrBC(ρp




DM

(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)DM

(ρ0

ABCRBrarrBC(ρ0AB)

)+ pDM

(ρ

pABCRBrarrBC(ρp

AB))

(443)


R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (444)






BrarrBC(ρAB))








sumx Mx = id





AB)))

ge supMisinM



ge supMisinM












12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

12












lim supirarrinfin


lim supirarrinfin

infρisinSa

Rai (ρ)


infρisinSai


Rai (Sai ) ge 0 (449)








13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

13







B otimesΠ cC)


C)ρABC)(52)

and




(53)



ABCcisinN
















C)+ 4h(εbc) (57)


ABC





14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

14



εbc le 2

radic1 minus F

(ρ

bcABC ρc

ABC

)2 le 2


tr(Πb

B otimesΠ cCρBC

)tr(Π c

CρC) (58)




(rankΠ c

C)+ 4h(εbc) (59)




ρbc minus tr(Π c

CρC)+ 1 (510)




(8εbc log(rankΠ c

C)+4h(εbc))


tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

) (511)



le ln(2)2

tr(Π c

CρC) (

8εbc log(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

le ln(2)2

(8εbc log

(rankΠ c

C)+ 4h(εbc)

)+ (

1 minus tr(Π c

CρC))

(512)



P(ρABC ρbc

ABC


ABC


B otimesΠ cCρBC

) (513)



ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ P

(RBrarrBC

(ρ

bcAB

)RBrarrBC(ρAB)

)le 2P

(ρABC ρbc

ABC

)+ P

(ρ

bcABCRBrarrBC

(ρ

bcAB

))

le P(ρ

bcABCRBrarrBC

(ρ

bcAB

))+ 2

radic1 minus tr

(Πb

B otimesΠ cCρBC

) (514)



15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

15



F(ρ

bcABCRBrarrBC

(ρ

bcAB

))2


1 minus tr(Πb

B otimesΠ cCρBC

))+ 4

radic1 minus tr

(Πb

B otimesΠ cCρBC

)


1 minus tr(Πb

B otimesΠ cCρBC

)

le(


2(

1 minus tr(Πb

B otimesΠ cCρBC

))14)2

(515)

This implies that




CρBC))14 (516)


R(ρbc) minus R(ρ)

le ln(2)2


CρC)) + 2radic


CρBC))14

= ξ bc (517)






assertion


lim supcrarrinfin

lim supbrarrinfin


lim supbrarrinfin

infρisinS

Rbc (ρ)


lim supbrarrinfin

infρisinS


= lim supcrarrinfin

lim supbrarrinfin

inf


minus ξ bc

= lim supcrarrinfin

lim supbrarrinfin

Rbc (Sbc)

ge 0 (518)





lim supkrarrinfin

Rk (S) ge 0 (519)


16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

16



R(S) ge 0 (520)




B andΠmB otimesΠm












C ) (522)





Similarly we obtain



B otimesΠmC )2






B )1ΠmB otimesΠm

C infin


B 1



mB )2 (525)




C )2 + 4radic



C )2 + 4radic




17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

17





ABC)


AB) ρmABC)


8+

radic2ε (527)

for

δm =radic

8


mB )2 +


mB otimesΠm

C )2 + 4radic


)12

(528)



2ε we have


radic2ε


(δm)2 + 8radic

2ε)2 (529)

As a result we find


radic8(2ε)14 (530)




C misinN



mB otimesΠm

C ) = 1







|tr(ΠmB otimesΠm


mB )))|



B ρBΠmB 1 (532)



C R(ΠmB ρABΠ

mB )) = tr(Πm

B otimesΠmC R(Πm

B ρBΠmB ))



mB 1





18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

18



ρB minusΠmB ρBΠ

mB 1 le 2


B ρBΠmB )2 = 2


(ρBΠm

B ρBΠmB

tr(ΠmB ρB)

)2

le 2radic










lim supkrarrinfin







Rm = limirarrinfin

[Rkmi ]m (539)




Rm (S) = infρisinS


limirarrinfin

[Rkm


infρisinS

[Rkm

i ]m (ρ)

= lim supirarrinfin

[Rkm


and hence





B= WΠm


b |b〉〈b|mB






19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

19



i iisinN such that


B leΠmprimeB Πm


BleΠmprime


ρmBCB

= Rm(ρBB)

= limirarrinfin


irarrinfin(Πm


i ]m+1(ΠmB ρBBΠ

mB )(Πm

B otimesΠmC )

= limirarrinfin

(ΠmB otimesΠm

C otimesΠmB



B)

= (ΠmB otimesΠm

C otimesΠmB


C otimesΠmB

)

= (ΠmB otimesΠm

C otimesΠmB

)ρm+1BCB

(ΠmB otimesΠm

C otimesΠmB

) (542)


B ΠmC leΠmprime

C and ΠmB

leΠmprimeB


ρmBCB

= (ΠmB otimesΠm

C otimesΠmB

)ρmprimeBCB

(ΠmB otimesΠm

C otimesΠmB

) (543)


F(ρmBCB

ρmprimeBCB



BCBΠm


B ρmprime

BCB)

ge tr(ρmprimeBCB

ΠmB otimesΠm

C otimesΠmB

) = tr(ρmBCB

) (544)


ρmBCB

minus ρmprimeBCB

1 le 2radic

tr(ρmprimeBCB

)2 minus F(ρmBCB

ρmprimeBCB

)2 le 2radic

tr(ρmprimeBCB

)2 minus tr(ρmBCB

)2 (545)






tr(ρmBCB

) le tr(ρmprimeBCB





BCB) = lim


tr([Rkmi ]m(ρBB))




irarrinfinεkm

i



radic1 minus tr(Πm

B ρB)2 = 1 (547)







20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

20



mrarrinfin ρmBCB

(548)





BCB) = lim



BrarrBC)(ρBB)

= limmrarrinfin(Rm


BrarrBC(ρAB) (549)




R(S) = infρisinS

R(ρ) = infρisinS







I(A C|B)ρ (61)





BCρ12BC (62)




21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

21




B UBXBUdaggerBρ


BC VdaggerBC (63)



I(A C|B)ρ (64)



ρp





B UBXBUdaggerBρ


BC VdaggerBC (66)



F(ρ

p

AABCRBrarrBC(ρp

AAB))

ge 1 minus ln(2)2

I(AA C|B)ρp (67)




AB)))

le pln(2)

2I(A C|B)ρ (68)











pminus δ le ln(2)

2I(A C|B)ρ (69)


p ge δ






22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


23



















BC (71)



24
















25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

22












BC = ρBC (611)



minus12B ρ

12BC VBCVdagger

BCρ12BC ρ


Bρminus12B ρBρ


BUB (612)





BCρ12BC (613)





I(A C|B)ρ (614)





we claim

F(ρ

p

AABCRBrarrBC(ρp

AAB))

= (1 minus p)F(ρ0

ABCRBrarrBC(ρ0AB)




RBrarrBC(ρp






R(ρp) = (1 minus p)R(ρ0) + pR(ρ) (618)


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BC (71)



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Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

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BC (71)



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Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

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26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

25




























26






















Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

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Math 2 209ndash218 (doi101007BF02825637)


Introduction

Main result

Applications













Discussion

References

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