Towards a one shot entanglement theory
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Transcript of Towards a one shot entanglement theory
Towards a one-s hot entanglement theory
Francesco Buscemi and Nilanjana Datta
\Beyond i.i.d. in information theory,"University of Cambridge, 9 January 2013
Part One:Introduction
Resource theory of (bipartite) entanglement
Entanglement is useful (quantum information processing) but expensive(difficult to establish and fragile to preserve)
+ study of entanglement as a resource
raw resources: bipartite quantum systems (in pure and/or mixedstates)
processing: local operations and classical communication (LOCC).(Why? Operational paradigm of “distant laboratories.”)
standard currency: the singlet state |Ψ−〉 = |01〉−|10〉√2
. (Why? Perhaps
because teleportation and superdense coding both use the singlet.)
basic tasks: distillation (extraction of singlets from raw resources) anddilution (creation of generic bipartite states from singlets) by LOCC
Asymptotic manipulation of (bipartite) quantumcorrelations
ρAB ⊗ ρAB ⊗ · · · ⊗ ρAB︸ ︷︷ ︸Min
L∈LOCC7−−−−−−→ σA′B′ ⊗ · · ·σA′B′︸ ︷︷ ︸Nout
where A-systems belong to Alice, B-systems belong to Bob, and thetransformation L is LOCC between Alice and Bob
Jargon: Min copies of the initial state ρAB are diluted into Nout
copies of the target state σA′B′ ; equivalently, Nout copies of thetarget state σA′B′ are distilled from Min copies of the initial state ρAB
Task is optimized with respect to the resources created (optimaldistillation, N = N(Min)) or those consumed (optimal dilution,M = M(Nout))
Optimal rates are computed as limMin→∞N(Min)/Min (optimaldistillation rate) and limNout→∞M(Nout)/Nout (optimal dilutionrate)
Asymptotic entanglement distillation and dilution
Entanglement distillation
ρAB ⊗ · · · ⊗ ρAB︸ ︷︷ ︸Min
L∈LOCC7−−−−−−→ Ψ−A′B′ ⊗ · · ·Ψ−A′B′︸ ︷︷ ︸N(Min)
distillable entanglement: E∞D (ρAB) = limMin→∞N(Min)/Min
Entanglement dilution
Ψ−AB ⊗ · · · ⊗Ψ−AB︸ ︷︷ ︸M(Nout)
L∈LOCC7−−−−−−→ σA′B′ ⊗ · · ·σA′B′︸ ︷︷ ︸Nout
entanglement cost: E∞C (σA′B′) = limNout→∞M(Nout)/Nout
Criticisms to this approach
The asymptotic framework is operational but not practical, for two reasons:
asymptotic achievability (and often without knowing how fast thelimit is approached)
i.i.d. assumption: hardly satisfied in practical scenarios
A third remark: the asymptotic i.i.d. argument mixes information theoryand probability theory. As noticed by Han and Verdu, we’d like todistinguish what is information theory from what is probability theory.
The one-shot case
One-shot entanglement distillation:
ρABL∈LOCC7−−−−−−→ Ψ−A′B′ ⊗ · · ·Ψ−A′B′︸ ︷︷ ︸
Nmax(ρAB)
.
One-shot entanglement dilution:
Ψ−AB ⊗ · · · ⊗Ψ−AB︸ ︷︷ ︸Mmin(σA′B′ )
L∈LOCC7−−−−−−→ σA′B′ .
Correspondingly,
I one-shot distillable entanglement: E(1)D (ρAB) = Nmax(ρAB);
I one-shot entanglement cost: E(1)C (σA′B′) = Mmin(σA′B′)
Allowing for finite accuracy
Again, with an eye to practical implementations:
One-shot entanglement ε-distillation:
ρABL∈LOCC7−−−−−−→ ρA′B′
ε≈ Ψ−A′B′ ⊗ · · ·Ψ−A′B′︸ ︷︷ ︸Nmax(ρAB ;ε)
.
One-shot entanglement ε-dilution
Ψ−AB ⊗ · · · ⊗Ψ−AB︸ ︷︷ ︸Mmin(σA′B′ ;ε)
L∈LOCC7−−−−−−→ σA′B′ε≈ σA′B′ .
Correspondingly,
one-shot ε-distillable entanglement: E(1)D (ρAB; ε) = Nmax(ρAB; ε);
one-shot entanglement ε-cost: E(1)C (σA′B′ ; ε) = Mmin(σA′B′ ; ε)
Outline of the talk
one-shot distillable entanglement (pure state case)
generalized entropies: Smin and Smax
one-shot entanglement cost (pure state case)
overview of the mixed state case: asymptotic results
relative Renyi entropies and derived quantities
mixed state case: one-shot results
comparison and discussion
Part Two:The Strange Case of Pure States
Case study: pure bipartite states
|ψAB〉 L∈LOCC7−−−−−−→ |φA′B′〉True in this case (but grossly false in general):
all the properties of a pure bipartite state ψAB are determined by thelist of eigenvalues ~λψ of the reduced density matrix ψA = TrB[ψAB];
Lo and Popescu: the action of a general LOCC map on a pure statecan be also obtained by another one-way, one-round LOCC map;
Nielsen: there exists an LOCC transformation mapping ψAB intoφA′B′ if and only if ψA ≺ φA′ , i.e.,
∑ki=1 λ
↓iψ 6
∑ki=1 λ
↓iφ , for all k;
asymptotic reversibility (total ordering):E∞D (ψAB) = E∞C (ψAB) = S(ψA).
One-shot zero-error distillable entanglement: E(1)D (ψAB; 0)
Nielsen: given an initial pure state ψAB, a maximally entangled state ofrank R, i.e. R−1/2
∑Ri=1 |i〉|i〉, can be distilled if and only if
λmaxψ ≡ λ↓1ψ 6 R−1, λ↓1ψ + λ↓2ψ 6 2R−1, and so on.
+ A maximally entangled state of rank R =⌊
1λmaxψ
⌋can always be
distilled exactly, i.e.,
E(1)D (ψAB; 0) > log2
⌊1
λmaxψ
⌋.
Finite accuracy: E(1)D (ψAB; ε)
Consider the set of pure states B∗ε (ψAB) :={|ψAB〉 : ψAB
ε≈ ψAB}
+ A maximally entangled state of rank R =
⌊1
λmaxψ
⌋can always be
distilled up to an ε-error, i.e.,
E(1)D (ψAB; ε) > max
ψ∈B∗ε (ψ)log2
⌊1
λmaxψ
⌋.
Getting the right smoothing
With B∗ε (ψAB) :={|ψAB〉 : ψAB
ε≈ ψAB}
:
E(1)D (ψAB; ε) > max
ψ∈B∗ε (ψ)log2
⌊1
λmaxψ
⌋︸ ︷︷ ︸
f(ψAB ,ε)
≡ Sεmin(ψA)
Given a (mixed) state ρ, define the set of (mixed) states
Bε(ρ) :={ρ : ρ
ε≈ ρ}
. The smoothed min-entropy of ρ is defined as
(Renner) Sεmin(ρ) := maxρ∈Bε(ρ) [− log2 λmax(ρ)].
Smin is the one-shot distillable entanglement
A converse also holds:
Sεmin(ψA) 6 E(1)D (ψAB; ε) 6 Sε
′min(ψA)− log2(1− 2
√ε).
[ε′ = 2
54 ε
18
]
+ min-entropy of the reduced state ≈ one-shot distillable entanglement ofa pure bipartite state.
Smin is the one-shot distillable entanglement
Beside the inequality E(1)D (ψAB; ε) � Sεmin(ψA), a converse can also be
proved:
E(1)D (ψAB; ε) � Sεmin(ψA)− log2(1− 2
√ε).
This corroborates the idea that the min-entropy of the reduced state reallyis the natural quantity measuring the one-shot distillable entanglement ofa pure bipartite state.
Smin(ψA)(α=∞)
· · · S(ψA)(α=1)
· · · Smax(ψA)(α=0)
Figure: is Smax associated with anything?
Smax is the one-shot entanglement cost
Vidal, Jonathan, and Nielsen: a pure bipartite state ψAB can beobtained by LOCC from a maximally entangled state of rank R with aminimum error of ε = 1−∑R
i=1 λ↓iψ .
As a consequence, E(1)C (ψAB; 0) = log2 rankψA = Smax(ψA).
With finite accuracy:
E(1)C (ψAB; ε) ' Sεmax(ψA),
where Sεmax(ρ) = minρ∈Bε(ρ) Smax(ρ).
Summary of the pure state case
E(1)D (ψAB; ε) ' Sεmin(ψA) 6 Sεmax(ψA) ' E
(1)C (ψAB; ε)
↓ ↘ ↙ ↓
E∞D (ψAB) = S(ψA) = E∞C (ψAB)
where “F (ρ; ε)→ G(ρ)” means limε→0 limn→∞ 1nF (ρ⊗n; ε) = G(ρ)
+ asymptotic reversibility holds for pure states
One-shot irreversibility gap for pure states
Reversibility only holds asymptotically. Define the one-shot irreversibilitygap as
∆(ψAB; ε) : = E(1)C (ψAB; ε)− E(1)
D (ψAB; ε)
' Sεmax(ψA)− Sεmin(ψA)
This quantity is related with the communication cost C of transforming aninitial pure state ψiAB into a final state ψfA′B′ (Hayden and Winter, 2003):
2C > ∆(ψfA′B′ ; 0)−∆(ψiAB; 0).
+ “Increasing irreversibility requires communication.”
Part Three:The Complicated Case of Mixed
States(an overview)
Mixed state case: asymptotic i.i.d. results
Distillable entanglement and entanglement cost are naturally quantified bydifferent functions of ρAB (Hayden, Horodecki, Terhal, 2001; Devetak,Winter, 2005):
E∞D (ρAB) E∞C (ρAB)o o
IA→Bc (ρAB)pure states↪−−−−−−→ S(ρA)
pure states←−−−−−−↩ minE∑
i piS(ψiA)
where:
IA→Bc (ρAB) = S(ρB)− S(ρAB) = −H(ρAB|B): coherentinformation
minE∑
i piS(ψiA) is done over all pure-state ensemble decompositionsρAB =
∑i piψ
iAB: entanglement of formation EF (ρAB)
Relative entropies and derived quantities
All such entropic quantities are originated from a common parent
Relative entropy:
S(ρ‖σ) = Tr [ρ log2 ρ− ρ log2 σ]
1 S(ρ) := −Tr[ρ log2 ρ] = −S(ρ‖1)2 H(ρAB |B) := S(ρAB)− S(ρB) =−minσB S(ρAB‖1A ⊗ σB)
3 IA→Bc (ρAB) := −H(ρAB |B)
Relative Renyi entropy of order zero:
S0(ρ‖σ) = − log2 Tr [Πρ σ]
1 S0(ρ) := −S0(ρ‖1) = Smax(ρ)
2 H0(ρAB |B) :=−minσB S0(ρAB‖1A ⊗ σB)
3 IA→B0 (ρAB) := −H0(ρAB |B)
Technical remark: quasi-entropies
In our proofs we employed the notion of quasi-entropies (Petz, 1986)
SPα (ρ‖σ) =1
α− 1log2 Tr
[√Pρα√P σ1−α
],
defined for ρ, σ > 0, 0 6 P 6 1, and α ∈ (0,∞)/{1}.
In particular, we enjoyed working with
SP0 (ρ‖σ) = limα↘0
SPα (ρ‖σ) = − log2 Tr[√
PΠρ
√P σ
],
smoothing w.r.t. ρ or P , depending on the problem at hand.
Mixed state case: one-shot results
Keeping in ming the asymptotic i.i.d. case:
E∞D (ρAB) E∞C (ρAB)o o
IA→Bc (ρAB)pure↪−−→ S(ρA)
pure←−−↩ minE∑
i piS(ψiA)‖
minEH(ρRA|R)
Here are the one-shot analogues:
E(1)D (ρAB; ε) E
(1)C (ρAB; ε)
o oIA→B0,ε (ρAB)
pure↪−−→ Sεmin(ρA) 6 Sεmax(ρA)
pure←−−↩ minEHε0(ρRA|R)
where minEHε0(ρRA|R) is done over all cq-extensions
ρRAB =∑
i pi|i〉〈i|R ⊗ ψiAB, such that TrR[ρRAB] = ρAB
A by-product worth noticing
Since E∞C (ρAB) = limε→0 limn→∞ 1nE
(1)C (ρ⊗nAB; ε), from the previous slide:
minEHε
0(ρRA|R)limε→0
1n
limn→∞−−−−−−−−−−−→ minEH(ρRA|R)
Both well-known guests of the zoo of entanglement measures:
minEH(ρRA|R) is the entanglement of formation (Bennett et al,1996) EF (ρAB) = minE
∑i piS(ψiA)
minEH0(ρRA|R) is the logarithm of the generalized Schmidt rank(Terhal, Horodecki, 2000) Esr(ρAB) = log2 minE maxi rankψiA
By introducing a smoothed Schmidt rank as follows:
Eεsr(ρAB) := minρAB∈Bε(ρAB)
Esr(ρAB),
we have implicitly proved that
limε→0
limn→∞
1
nEεsr(ρ
⊗nAB) = lim
n→∞1
nEF (ρ⊗nAB).
Conclusions and open questions
mix- and max-entropies naturally arise also in one-shot entanglementtheory
pleasant formal analogy with the asymptotic i.i.d. case: just replaceS(ρ‖σ) by S0(ρ‖σ) (but, first, find the right expression to replace!)
sometimes, the one-shot analysis uncovers new relations betweenknown functions (e.g. the regularized entanglement of formationequals the smoothed-and-regularized log-Schmidt rank)
increasing irreversibility requires communication: what about mixedstates?
other operational paradigms: SEPP done (Fernando and Nila); whatabout LOSR?
one-shot squashed entanglement: one-shot quantum conditionalmutual information?
La Fine.