Trading Resources in Quantum Shannon Theory

Trading resources in quantum Shannon theory

Mark M. Wilde

Hearne Institute for Theoretical Physics,Department of Physics and Astronomy,

Center for Computation and Technology,Louisiana State University,

Baton Rouge, Louisiana, USA

[email protected]

Based on arXiv:1206.4886, 1105.0119, 1004.0458, 1001.1732, 0901.3038, 0811.4227 (withBradler, Guha, Hayden, Hsieh, Touchette) and Chapter 25 of arXiv:1106.1445

NICT, December 15, 2015, Koganei, Tokyo 184-8795, Japan

Mark M. Wilde (LSU) 1 / 41

Main message

Question: What are the net rates at which a sender and receiver cangenerate classical communication, quantum communication, andentanglement by using a channel many times?

Many special cases are known, such as the classical capacity theorem[Hol98, SW97], quantum capacity theorem[Sch96, SN96, BNS98, BKN00, Llo97, Sho02, Dev05], and theentanglement-assisted classical capacity theorem [BSST02]

A priori, this question might seem challenging, but there is asurprisingly simple answer for several channels of interest:

Just combine a single protocol with teleportation, super-dense coding,and entanglement distribution


Background — resources

Resources [Ben04, DHW04, DHW08]

Let [c → c] denote a noiseless classical bit channel from Alice(sender) to Bob (receiver), which performs the following mapping ona qubit density operator

ρ =

[ρ00 ρ01

ρ10 ρ11

]→[ρ00 00 ρ11

]Let [q → q] denote a noiseless quantum bit channel from Alice toBob, which perfectly preserves a qubit density operator.

Let [qq] denote a noiseless ebit shared between Alice and Bob, whichis a maximally entangled state |Φ+〉AB = (|00〉AB + |11〉AB)/

√2.

Entanglement distribution, super-dense coding, and teleportation arenon-trivial protocols for combining these resources


Entanglement distribution

A’

B

id

|0〉

|0〉

A

A’→B

H

Alice performs local operations (the Hadamard and CNOT) andconsumes one use of a noiseless qubit channel to generate onenoiseless ebit |Φ+〉AB shared with Bob.

Resource inequality: [q → q] ≥ [qq]


Super-dense coding [BW92]

X Z

Bell Measurement

Conditional Operations

QubitChannel

x1x2

x1x2

|Ф 〉+AB

Alice and Bob share an ebit. Alice would like to transmit two classicalbits x1x2 to Bob. She performs a Pauli rotation conditioned on x1x2

and sends her share of the ebit over a noiseless qubit channel. Bobthen performs a Bell measurement to get x1x2.

Resource inequality: [q → q] + [qq] ≥ 2[c → c]


Teleportation [BBC+93]

X Z

Bell Measurement

Conditional Operations

Two ClassicalChannels

|Ф 〉+AB

A’|ψ〉

B|ψ〉

Alice would like to transmit an arbitrary quantum state |ψ〉A′ to Bob.Alice and Bob share an ebit before the protocol begins. Alice can“teleport” her quantum state to Bob by consuming the entanglementand two uses of a noiseless classical bit channel.

Resource inequality: 2[c → c] + [qq] ≥ [q → q]


Combining protocols [HW10]

Think of each protocol as a rate triple (C ,Q,E )

Entanglement distribution is (0,−1, 1)

Super-dense coding is (2,−1,−1)

Teleportation is (−2, 1,−1)

All achievable rate triples are then given by

{(C ,Q,E ) = α(−2, 1,−1) + β(2,−1,−1) + γ(0,−1, 1) : α, β, γ ≥ 0}

Writing as a matrix equation, inverting, and applying constraintsα, β, γ ≥ 0 gives the following achievable rate region:

C + Q + E ≤ 0,

Q + E ≤ 0,

C + 2Q ≤ 0.


Unit resource capacity region [HW10]

1

0.5

0

-0.5

-1-1

01 2

0-2

ED

SD TP

(0,-1,1)

(0,0,0)

(2,-1,-1)(-2,1,-1)

E

Q C

The unit resource capacity region is C + Q + E ≤ 0, Q + E ≤ 0,C + 2Q ≤ 0 and is provably optimal.


Trading resources using a quantum channel

Main question: What net rates of classical communication, quantumcommunication, and entanglement generation can we achieve byusing a quantum channel N many times?

That is, what are the rates Cout, Qout, Eout, Cin, Qin, Ein ≥ 0achievable in the following resource inequality?

〈N〉+ Cin[c → c] + Qin[q → q] + Ein[qq]

≥ Cout[c → c] + Qout[q → q] + Eout[qq]

The union of all achievable rate triples(Cout − Cin,Qout − Qin,Eout − Ein) is called the quantum dynamiccapacity region.


Trading resources using a quantum channel

TA

TB

A1

R

A’

A’

A’

B

B

B

BB

BB1

Alice

Bob

Reference t tf

D

NN

N

EM

SA

SBL

A2

M

Figure: The most general protocol for generating classical communication,quantum communication, and entanglement with the help of the same respectiveresources and many uses of a quantum channel.


Background — entropies

The optimal rates are expressed in terms of entropies, which wereview briefly

Given a density operator σ, the quantum entropy is defined asH(σ) = −Tr{σ log σ}.Given a bipartite density operator ρAB , the quantum mutualinformation is defined as

I (A;B)ρ = H(A)ρ + H(B)ρ − H(AB)ρ

The coherent information I (A〉B)ρ is defined as

I (A〉B)ρ = H(B)ρ − H(AB)ρ

Given a tripartite density operator ρABC , the conditional mutualinformation is defined as

I (A;B|C )ρ = H(AC )ρ + H(BC )ρ − H(C )ρ − H(ABC )ρ


Quantum dynamic capacity theorem (setup) [WH12]

Define the state-dependent region C(1)CQE,σ(N ) as the set of all rates

C , Q, and E , such that

C + 2Q ≤ I (AX ;B)σ,

Q + E ≤ I (A〉BX )σ,

C + Q + E ≤ I (X ;B)σ + I (A〉BX )σ.

The above entropic quantities are with respect to a classical–quantumstate σXAB , where

σXAB ≡∑x

pX (x)|x〉〈x |X ⊗NA′→B(φxAA′),

and the states φxAA′ are pure.


Quantum dynamic capacity theorem (statement) [WH12]

Define C(1)CQE(N ) as the union of the state-dependent regions

C(1)CQE,σ(N ):

C(1)CQE(N ) ≡

⋃σ

C(1)CQE,σ(N ).

Then the quantum dynamic capacity region CCQE(N ) of a channel Nis equal to the following expression:

CCQE(N ) =∞⋃k=1

1

kC(1)

CQE(N⊗k),

where the overbar indicates the closure of a set.

It is implicit that one should consider states on A′k instead of A′

when taking the regularization.


Example: Qubit dephasing channel

Take the channel to be the qubit dephasing channelN (ρ) = (1− p)ρ+ pZρZ with dephasing parameter p = 0.2.

Take the input state as

σXAA′ ≡ 1

2(|0〉〈0|X ⊗ φ0

AA′ + |1〉〈1|X ⊗ φ1AA′),

where ∣∣φ0⟩AA′ ≡

√1/4|00〉AA′ +

√3/4|11〉AA′ ,∣∣φ1

⟩AA′ ≡

√3/4|00〉AA′ +

√1/4|11〉AA′ .

The state σXAB resulting from the channel is NA′→B(σXAA′)


Example: Qubit dephasing channel (ctd.)

EAC

CEF-SD-EDCEQ LSD

CEFEAQ

CEF-TP

Entanglement consum

ption rate

Quantum communicationrate Classical communication rate

I

II

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.100

0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 1.5

10.5

0

III

Figure: An example of the state-dependent achievable region C(1)CQE σ(N )

corresponding to a state σXABE that arises from a qubit dephasing channel withdephasing parameter p = 0.2. The figure depicts the octant corresponding to theconsumption of entanglement and the generation of classical and quantumcommunication.


Direct part of the quantum dynamic capacity theorem

Entanglement-assisted classical and quantum communication

There is a protocol that implements the following resource inequality:

〈N〉+1

2I (A;E |X )ρ [qq] ≥ 1

2I (A;B|X )ρ [q → q] + I (X ;B)ρ [c → c]

where ρXABE is a state of the following form:

ρXABE ≡∑x

pX (x)|x〉〈x |X ⊗ UNA′→BE (ϕxAA′),

the states ϕxAA′ are pure, and UNA′→BE is an isometric extension of the

channel NA′→B .

Combine this with the unit protocols of teleportation, super-densecoding, and entanglement distribution



Combining the protocols gives the following set of achievable rates:CQE

=

0 2 −2−1 −1 11 −1 −1

αβγ

+

I (X ;B)σ12 I (A;B|X )σ−1

2 I (A;E |X )σ

,where α, β, γ ≥ 0.

Inverting the matrix equation, applying the constraints α, β, γ ≥ 0,and using entropy identities gives the following region:

C + 2Q ≤ I (AX ;B)σ,

Q + E ≤ I (A〉BX )σ,

C + Q + E ≤ I (X ;B)σ + I (A〉BX )σ,

which establishes the achievability part.



How to achieve the following resource inequality?

〈N〉+1

2I (A;E |X )ρ [qq] ≥ 1

2I (A;B|X )ρ [q → q] + I (X ;B)ρ [c → c]

Tools for achievability part [Wil15, Chapter 25]

HSW classical capacity theorem [Hol98, SW97]

Entanglement-assisted classical capacity theorem [BSST02] (see also[HDW08])

Modification of a classical trick called “superposition coding” [Sho04]

Another trick called coherent communication [Har04, DHW08]


HSW theorem (constant-composition variant)

Fix an ensemble {pX (x), ρxA} and set σxB ≡ NA→B(ρxA).

Now select a typical type class Tt , which is a set of all the sequencesxn with

1 the same empirical distribution t(x)2 t(x) deviates from the distribution pX (x) by no more than δ > 0

All the sequences in the same type class are related to one another bya permutation, and all of them are strongly typical

Select a code at random by picking all of the codewordsindependently and uniformly at random from the typical type class

We can then conclude the existence of a codebook {xn(m)}m∈M anda decoding POVM {Λm

Bn}m∈M such that M≈ 2nI (X ;B) and

Tr{

ΛmBnN⊗n

(ρxn(m)An

)}≥ 1− ε ∀m ∈M


Entanglement-assisted coding (simple version)

Allow Alice and Bob to share a maximally entangled state |Φ〉AB

They then induce the following ensemble by Alice applying aHeisenberg–Weyl operator uniformly at random:{

d−2, (NA→B′ ⊗ idB) (Φx ,zAB)}.

where |Φx ,z〉AB = X (x)A Z (z)A |Φ〉AB . (This is the same ensemblefrom super-dense coding if N is the identity channel.)

By the HSW theorem and some entropy manipulations, we canconclude that the mutual information I (B ′;B)N (Φ) is an achievablerate.


Entanglement-assisted coding (general version)

Allow Alice and Bob to share many copies of a pure bipartite state

|ϕ〉AB ≡∑x

√pX (x)|x〉A|x〉B .

Much degeneracy in many copies of this state—can rewrite it as

|ϕ〉⊗nAB =∑xn

√pX n(xn)|xn〉An |xn〉Bn =

∑t

√p(t)|Φt〉AnBn

where |Φt〉AnBn is maximally entangled on a type class subspace t.

Take encoding unitary to have the form

U(s) ≡⊕t

(−1)bt V (xt , zt)

where V (xt , zt) is a Heisenberg–Weyl operator for a type classsubspace t and s = ((xt , zt , bt)t).


Entanglement-assisted coding (general version)

Random coding: pick encoding unitaries U(s) uniformly at random

The entanglement-assisted quantum codewords

|ϕm〉AnBn = (UAn(s(m))⊗ IBn) |ϕ〉⊗nAB

have the following interesting property:

|ϕm〉AnBn =(IAn ⊗ UT

Bn(s(m)))|ϕ〉⊗nAB ,

which allows us to conclude that the reduced state on the channelinput is the same for all codewords:

TrBn {|ϕm〉〈ϕm|AnBn} = ϕ⊗nA

(privacy without access to Bob’s share of the entanglement)

Can achieve the mutual information rate I (B ′;B)N (ϕ)


“Superposition coding” [Sho04]

“Layer” an HSW code “on top of” several EA codes:

l3

l2

Ul1

l1

Ul2

l2

Ul3

l3

|φ1〉

|φ1〉

|φ1〉

|φ2〉

|φ2〉

|φ3〉

|φ3〉

N

N

N

N

N

N

N

m m

l1

This achieves the following resource inequality:

〈N〉+ H(A|X )ρ [qq] ≥ I (A;B|X )ρ [c → c] + I (X ;B)ρ [c → c]

where ρXAB ≡∑

x pX (x)|x〉〈x |X ⊗NA′→B(ϕxAA′).


Upgrading with coherent communication [Har04, DHW08]

It is possible to “upgrade” the classical bits transmitted by theentanglement-assisted codes to “coherent bits”, because they areprivate from the environment of the channel [DHW08]

We can then use a trick called the coherent communication identity[Har04] to conclude that the desired resource inequality is achievable:

〈N〉+1

2I (A;E |X )ρ [qq] ≥ 1

2I (A;B|X )ρ [q → q] + I (X ;B)ρ [c → c]

where ρXABE ≡∑

x pX (x)|x〉〈x |X ⊗ UNA′→BE (ϕxAA′).


Converse part

Consider the most general protocol:

TA

TB

A1

R

A’

A’

A’

B

B

B

BB

BB1

Alice

Bob

Reference t tf

D

NN

N

EM

SA

SBL

A2

M

Make use of quantum data processing and dimension bounds forinformation quantities


Computing the boundary of the region [WH12]

Let ~w ≡ (wC ,wQ ,wE ) ∈ R3 be a weight vector, ~R ≡ (C ,Q,E ) a ratevector, and E ≡ {pX (x), φxAA′} an ensemble.

Can phrase the task of computing the boundary of the single-copycapacity region as an optimization problem:

P∗(~w) ≡ sup~R,E

~w · ~R

subject to C + 2Q ≤ I (AX ;B)σ,

Q + E ≤ I (A〉BX )σ,

C + Q + E ≤ I (X ;B)σ + I (A〉BX )σ,

where the optimization is with respect to all rate vectors ~R andensembles E , with σXAB a state of the previously given form.


Quantum dynamic capacity formula [WH12]

By linear programming duality, if P∗(~w) <∞, then the optimizationproblem is equivalent to computing the quantum dynamic capacityformula, defined as

D~λ(N ) ≡ maxσλ1I (AX ;B)σ+λ2I (A〉BX )σ+λ3 [I (X ;B)σ + I (A〉BX )σ] ,

where σXAB is a state of the previously given form and~λ ≡ (λ1, λ2, λ3) is a vector of Lagrange multipliers such thatλ1, λ2, λ3 ≥ 0.

Suppose for a given channel N that D~λ(N⊗n) = nD~λ(N ) ∀n ≥ 1

and ~λ � 0. Then the computation of the boundary simplifiessignificantly. This happens for a number of important channels.


Example: Quantum erasure channel

Erasure channel is defined as follows:

N ε(ρ) = (1− ε) ρ+ ε|e〉〈e|,

where ρ is a d-dimensional input state, |e〉 is an erasure flag stateorthogonal to all inputs (so that the output space has dimensiond + 1), and ε ∈ [0, 1] is the erasure probability.

Let N ε be a quantum erasure channel with ε ∈ [0, 1/2]. Then thequantum dynamic capacity region CCQE(N ε) is equal to the union ofthe following regions, obtained by varying λ ∈ [0, 1]:

C + 2Q ≤ (1− ε) (1 + λ) log d ,

Q + E ≤ (1− 2ε)λ log d ,

C + Q + E ≤ (1− ε− ελ) log d .


Example: Quantum erasure channel

−1.5−1−0.500.511.522.5

−1−0.5

00.5

11.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Classical communication rateQuantum communication rate

Enta

ngle

men

t con

sum

ptio

n ra

te SDSD

EDED

TP

CEF curve

Figure: The quantum dynamic capacity region for the (qubit) quantum erasurechannel with ε = 1/4. The plot demonstrates that time-sharing is optimal.



The dynamic capacity region CCQE(∆p) of a dephasing channel withdephasing parameter p ∈ [0, 1] is the set of all C , Q, and E such that

C + 2Q ≤ 1 + h2(ν)− h2(γ(ν, p)),

Q + E ≤ h2(ν)− h2(γ(ν, p)),

C + Q + E ≤ 1− h2(γ(ν, p)),

where ν ∈ [0, 1/2], h2 is the binary entropy function, and

γ(ν, p) ≡ 1

2+

1

2

√1− 16 · p

2

(1− p

2

)ν(1− ν).



−2−10123

−10

12

−1

−0.5

0

0.5

1

1.5

Classical communication rateQuantum communication rate

Enta

ngle

men

t con

sum

ptio

n ra

te

ED

ED

TPSD

SD

CEF curve

Figure: A plot of the dynamic capacity region for a qubit dephasing channel withdephasing parameter p = 0.2. Slight improvement over time-sharing.


Example: Pure-loss bosonic channel

Pure-loss channel is defined from the following input-output relation:

a → b =√η a +

√1− η e,

e → e ′ = −√

1− η a +√η e,

where a is the input annihilation operator for the sender, e is theinput annihilation operator for the environment, and η ∈ [0, 1] is thetransmissivity of the channel.

Place a photon number constraint on the input mode to the channel,such that the mean number of photons at the input cannot be greaterthan NS ∈ [0,∞).


Example: Pure-loss bosonic channel [WHG12]

Build trade-off codes from an ensemble of the following form:{p(1−λ)NS

(α),DA′(α)|ψTMS(λ)〉AA′},

where α ∈ C,

p(1−λ)NS(α) ≡ 1

π (1− λ)NSexp

{− |α|2 / [(1− λ)NS ]

},

λ ∈ [0, 1] is a photon-number-sharing parameter, DA′(α) is a“displacement” unitary operator acting on system A′, and |ψTMS(λ)〉AA′ isa “two-mode squeezed” (TMS) state:

|ψTMS(λ)〉AA′ ≡∞∑n=0

√[λNS ]n

[λNS + 1]n+1|n〉A|n〉A′ ,



The quantum dynamic capacity region for a pure-loss bosonic channel withtransmissivity η ≥ 1/2 is the union of regions of the form:

C + 2Q ≤ g(λNS) + g(ηNS)− g((1− η)λNS),

Q + E ≤ g(ηλNS)− g((1− η)λNS),

C + Q + E ≤ g(ηNS)− g((1− η)λNS),

where λ ∈ [0, 1] is a photon-number-sharing parameter and g(N) is theentropy of a thermal state with mean photon number N. (This holdsprovided that an unsolved minimum-output entropy conjecture is true.)The region is still achievable if η < 1/2.



0 2 4 6 8

Q (q

ubits

/ ch

anne

l use

)

9 10 11

Trade−off codingTime−sharing

(a) (b)

1.6

1.2

0.8

0.4

0

C (cbits / channel use)

E (e

bits

/ ch

anne

l use

) 8

6

4

2

0

C (cbits / channel use)

η = 3/4N = 200S

Figure: Suppose channel transmits on average 3/4 of the photons to the receiver,while losing the other 1/4 en route. Take mean photon budget of about200 photons per channel use at the transmitter. (a) classical–quantum trade-off,(b) classical comm. with rate-limited entanglement consumption. Big gains overtime-sharing.


Conclusion

Summary

The quantum dynamic capacity theorem characterizes the net rates atwhich a sender and a receiver can generate classical communication,quantum communication, and entanglement by using a quantumchannel many times

The region simplifies for several channels of interest

Open questions

Is there a simple characterization for distillation tasks? For progress,see [HW10]

Can we sharpen the theorem? Strong converse bounds, errorexponents, finite-length, second-order, etc.

What about channel simulation tasks? (see, e.g., [BDH+14])


References I

[BBC+93] Charles H. Bennett, Gilles Brassard, Claude Crepeau, Richard Jozsa, AsherPeres, and William K. Wootters. Teleporting an unknown quantum state viadual classical and Einstein-Podolsky-Rosen channels. Physical ReviewLetters, 70(13):1895–1899, March 1993.

[BDH+14] Charles H. Bennett, Igor Devetak, Aram W. Harrow, Peter W. Shor, andAndreas Winter. The quantum reverse Shannon theorem and resourcetradeoffs for simulating quantum channels. IEEE Transactions onInformation Theory, 60(5):2926–2959, May 2014. arXiv:0912.5537.

[Ben04] Charles H. Bennett. A resource-based view of quantum information.Quantum Information and Computation, 4:460–466, December 2004.

[BKN00] Howard Barnum, Emanuel Knill, and Michael A. Nielsen. On quantumfidelities and channel capacities. IEEE Transactions on Information Theory,46(4):1317–1329, July 2000. arXiv:quant-ph/9809010.

[BNS98] Howard Barnum, M. A. Nielsen, and Benjamin Schumacher. Informationtransmission through a noisy quantum channel. Physical Review A,57(6):4153–4175, June 1998.


References II

[BSST02] Charles H. Bennett, Peter W. Shor, John A. Smolin, and Ashish V.Thapliyal. Entanglement-assisted capacity of a quantum channel and thereverse Shannon theorem. IEEE Transactions on Information Theory,48(10):2637–2655, October 2002. arXiv:quant-ph/0106052.

[BW92] Charles H. Bennett and Stephen J. Wiesner. Communication via one- andtwo-particle operators on Einstein-Podolsky-Rosen states. Physical ReviewLetters, 69(20):2881–2884, November 1992.

[Dev05] Igor Devetak. The private classical capacity and quantum capacity of aquantum channel. IEEE Transactions on Information Theory, 51(1):44–55,January 2005. arXiv:quant-ph/0304127.

[DHW04] Igor Devetak, Aram W. Harrow, and Andreas Winter. A family of quantumprotocols. Physical Review Letters, 93(23):239503, December 2004.arXiv:quant-ph/0308044.

[DHW08] Igor Devetak, Aram W. Harrow, and Andreas Winter. A resource frameworkfor quantum Shannon theory. IEEE Transactions on Information Theory,54(10):4587–4618, October 2008. arXiv:quant-ph/0512015.


References III

[Har04] Aram Harrow. Coherent communication of classical messages. PhysicalReview Letters, 92(9):097902, March 2004. arXiv:quant-ph/0307091.

[HDW08] Min-Hsiu Hsieh, Igor Devetak, and Andreas Winter. Entanglement-assistedcapacity of quantum multiple-access channels. IEEE Transactions onInformation Theory, 54(7):3078–3090, July 2008. arXiv:quant-ph/0511228.

[Hol98] Alexander S. Holevo. The capacity of the quantum channel with generalsignal states. IEEE Transactions on Information Theory, 44(1):269–273,January 1998. arXiv:quant-ph/9611023.

[HW10] Min-Hsiu Hsieh and Mark M. Wilde. Trading classical communication,quantum communication, and entanglement in quantum Shannon theory.IEEE Transactions on Information Theory, 56(9):4705–4730, September2010. arXiv:0901.3038.

[Llo97] Seth Lloyd. Capacity of the noisy quantum channel. Physical Review A,55(3):1613–1622, March 1997. arXiv:quant-ph/9604015.

[Sch96] Benjamin Schumacher. Sending entanglement through noisy quantumchannels. Physical Review A, 54(4):2614–2628, October 1996.


References IV

[Sho02] Peter W. Shor. The quantum channel capacity and coherent information. InLecture Notes, MSRI Workshop on Quantum Computation, 2002.

[Sho04] Peter W. Shor. Quantum Information, Statistics, Probability (Dedicated toA. S. Holevo on the occasion of his 60th Birthday): The classical capacityachievable by a quantum channel assisted by limited entanglement. RintonPress, Inc., 2004. arXiv:quant-ph/0402129.

[SN96] Benjamin Schumacher and Michael A. Nielsen. Quantum data processingand error correction. Physical Review A, 54(4):2629–2635, October 1996.arXiv:quant-ph/9604022.

[SW97] Benjamin Schumacher and Michael D. Westmoreland. Sending classicalinformation via noisy quantum channels. Physical Review A, 56(1):131–138,July 1997.

[WH12] Mark M. Wilde and Min-Hsiu Hsieh. The quantum dynamic capacity formulaof a quantum channel. Quantum Information Processing, 11(6):1431–1463,December 2012. arXiv:1004.0458.


References V

[WHG12] Mark M. Wilde, Patrick Hayden, and Saikat Guha. Information trade-offs foroptical quantum communication. Physical Review Letters, 108(14):140501,April 2012. arXiv:1105.0119.

[Wil15] Mark M. Wilde. From Classical to Quantum Shannon Theory. 2015.arXiv:1106.1445 (preprint for a 2nd edition).


Trading Resources in Quantum Shannon Theory

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