Strong Converse Rates for Quantum Communication fileStrong Converse Rates for Quantum Communication...

Strong Converse Rates for QuantumCommunication

Marco Tomamichel1, Mark M. Wilde2, Andreas Winter3

1School of Physics, The University of Sydney2Center for Computation and Technology, Dept. of Physics & Astronomy,

Louisiana State University3ICREA & Fısica Teorica, Informacio i Fenomens Quantics,

Universitat Autonoma de Barcelona

ISIT 2015, Hong Kong

Introduction Coding for Quantum Channels Main Result and Proof Dephasing Channels Conclusion

Quantum Information Theory

• Information theory is concerned with establishing fundamentallimits for information processing tasks.

• The most precise modeling of physical information carriers(known today) uses the formalism of quantum mechanics.

Source: Wikipedia

• Many fundamental questions remain unanswered, for example:

1 No single-letter formula for point-to-point channel capacity.2 Strong converse for memoryless channels unknown.

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Quantum Primer: States

classical world quantum world

alphabet: X = {0, 1} Hilbert space: X = C2 — “qubit”orthonormal basis: |0〉, |1〉 ∈ X.

pmf: PX(x) = px quantum state: ρX =

(p0 00 p1

)generally: ρX ∈ S(X), linear operator on Xwith ρX ≥ 0 and tr(ρX) = 1.

t t

t t

t

|1〉

|0〉

t t|−〉 |+〉

|±〉 = 1√2(|0〉 ± |1〉)

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Quantum Primer: Correlation vs. Entanglement

• Relative entropy: D(ρ‖σ) = tr(ρ(log ρ− log σ)

).

• Entropy: H(A)ρ = −D(ρA‖1A) = − tr(ρ log ρ).

• Cond. entropy: H(A|B)ρ = −D(ρAB‖1A ⊗ ρB) = H(AB)ρ −H(B)ρ.


correlated pmf: entangled quantum state:

PXX′(x, x′) =

{px x = x′

0 else|φ〉XX′ =

√p0|0〉 ⊗ |0〉+

√p1|1〉 ⊗ |1〉

φXX′ = |φ〉〈φ|XX′ =

p0 0 0

√p0p1

0 0 0 00 0 0 0√p0p1 0 0 p1

not extremal extremal (pure, rank one)

entropy: H(X)P H(X)φ = H(X)Pcond. entropy: H(X|X ′)P = 0 H(X|X ′)φ = −H(X)P

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).





PXX′(x, x′) =

{px x = x′

0 else|φ〉XX′ =

√p0|0〉 ⊗ |0〉+

√p1|1〉 ⊗ |1〉

φXX′ = |φ〉〈φ|XX′ =

p0 0 0

√p0p1

0 0 0 00 0 0 0√p0p1 0 0 p1

not extremal extremal (pure, rank one) not extremal (mixed)


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).





PXX′(x, x′) =

{px x = x′

0 else|φ〉XX′ =

√p0|0〉 ⊗ |0〉+

√p1|1〉 ⊗ |1〉

φXX′ = |φ〉〈φ|XX′ =

p0 0 0

√p0p1

0 0 0 00 0 0 0√p0p1 0 0 p1

not extremal extremal (pure, rank one)


6 / 22


Quantum Coding: Channels

• Quantum channel: completely positive trace-preservinglinear map N ≡ NA→B from (states on) A to (states on) B.

NA B

Assume A and B are finite-dimensional.

• The channel is memoryless:

NAn Bn

NA2 B2

NA1 B1

≡ N⊗nAn Bn

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Quantum Coding: Encoder and Decoder

• Entanglement transmission code (for N⊗n):

Cn = {dn, En,Dn) .

1 code size dn:• M , M ′, M ′′ of dimension dn.• maximally entangled state

|φ〉MM′ =1√dn

dn∑i=1

|i〉M⊗|i〉M′ .

2 encoder En: quantum channelfrom M ′ to An.

3 decoder Dn: quantum channelfrom Bn to M ′′.

|φ〉

M ′

M

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Cn = {dn, En,Dn) .


|φ〉MM′ =1√dn

dn∑i=1

|i〉M⊗|i〉M′ .



M ′

En

An

A2

A1

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Cn = {dn, En,Dn) .


|φ〉MM′ =1√dn

dn∑i=1

|i〉M⊗|i〉M′ .



M ′′

Dn

Bn

B2

B1

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Quantum Coding: Entanglement Fidelity

N

N

N|φ〉

M ′

M

M ′′

En Dn

An

A2

A1

Bn

B2

B1

• Fidelity with maximally entangled state:

F (Cn, N⊗n) = tr(

(Dn ◦ N⊗n ◦ En)(φMM ′)φMM ′′).

• Corresponds to Pr[M = M ′′] classically.

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Quantum Capacity

• A triple (R,n, ε) is achievable on N if ∃ Cn with

1

nlog dn ≥ R, and F (CnN⊗n) ≥ 1− ε .

• Boundary of (non-asymptotic) achievable region:

R(n; ε,N ) := max{R : (R,n, ε) is achievable on N

}.

• The quantum capacity, Q(N ), is the rate at which qubits canbe transmitted with fidelity approaching one asymptotically.

Qε(N ) := limn→∞

R(n; ε,N ), ε ∈ (0, 1)

Q(N ) := limε→0

Qε(N ) .

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Quantum Capacity Theorem

• Barnum, Nielsen and Schumacher (1996-2000) as well asLloyd, Shor and Devetak (1997-2005) established

Q(N ) = lim`→∞

1

`Ic

(N⊗`

), where

Ic(N ) = maxρA{−H(A|B)ω} , ωAB = NA′→B(ψρA′A)

• This result is unsatisfactory for many reasons:

1 It is not a single-letter formula.2 The limit `→∞ is necessary in general (Cubitt et al.’14).3 It cannot be calculated except for e.g. degradable channels

which satisfy Ic(N⊗n

)= nIc(N ).

4 We do not know anything about Qε(N ).

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Strong Converse Property

• A quantum channel satisfies the strong converse property iff

Qε(N ) = Q(N ) for all ε ∈ (0, 1) .

0.40 0.45 0.50 0.55 0.60 0.650.0

0.2

0.4

0.6

0.8

1.0

n→∞

rate, R

toleratederror,ε

HHYachievable region

capacity ?

• A strong converse rate is a uniform upper bound on Qε.

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State of the Art

• Until this work, the strong converse property could onlybe established for some channels with trivial capacity.

• Morgan and Winter showed that degradable quantumchannels satisfy a “pretty strong” converse:

Qε(N ) = Q(N ) for all ε ∈(

0,1

2

)

(Extending their proof to all ε ∈ (0, 1) appears difficult.)

• Strong converse rates are known, for example theentanglement-assisted capacity established via channelsimulation by Bennett et al. and Berta et al.

• However, they are not tight except for trivial channels.

A lot of (fundamental) work still needs to be done!

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Result 1: Rains Entropy is Strong Converse Rate

• The Rains relative entropy of the channel is defined as

R(N ) := maxρA

minσAB∈Rains(A:B)

D(NA′→B(ψρA′A)

∥∥σAB).

• A state σAB ∈ Rains(A : B) (cf. Rains’99) satisfies

tr(φABσAB

)≤ 1

d∀ maximally entangled φAB.

Result

For any channel N , communication at a rate exceeding R(N )implies (exponentially) vanishing fidelity.

• Key Idea: Consider correlations σAB that are useless forquantum communication. Classically:

C(W ) = maxPX

minQX ,QY

D(PX ×WY |X

∥∥QX ×QY).

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Step 1: Arimoto-Type (One-Shot) Converse Bound

• Consider C = {d, E ,D} for N with F (C,N ) ≥ 1− ε.

• Test if a state is φMM ′′ , or not:

T (·) = p|0〉〈0|+ (1− p)|1〉〈1|, p = tr(φMM ′′ ·).

• Let ρAM = E(φMM ′). Due to data-processing, we have

Dα(N (ρAM )‖σBM ) ≥ Dα

(T ◦ D ◦ N (ρAM )‖T ◦ D(σRB)

)

≥ log d+α

α− 1log(1− ε),

for Renyi divergence with α > 1

• Sandwiched Renyi divergence (Lennert et al., Wilde et al.’13):

Dα(ρ‖σ) =1

α− 1log tr

((σ

1−α2α ρσ

1−α2α

)α).

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Step 2: Asymptotics

• Minimizing σAB ∈ Rains(A : B) and optimizing over codes:

Lemma

We have the following one-shot converse:

R(1; ε,N ) ≤ maxρA

minσAB

Dα(NA′→B(ψρAA′)‖σAB) +α log 1

1−εα− 1

• This yields an upper bound on the ε-capacity:

Qε(N ) ≤ lim supn→∞

1

nmaxρAn

minσAnBn

Dα

(N⊗n(ψρAnA′n)

∥∥σAnBn)

︸︷︷︸Rα(N⊗n)

.

• It remains to show that Rα(N ) satisfies an asymptoticsub-additivity property, i.e. Rα(N⊗n) ≤ nRα(N ) + o(n).

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Step 3: Covariant Channels and Permutations

• Covariance group of the channel N : Group G with unitaryrepresentations UA and VB such that

NA→B(UA(g)( · )U †A(g)) = VB(g)NA→B( · )V †B(g) ∀g ∈ G

• We show that:

Rα(N ) = maxρA

minσAB

Dα

(NA′→B(ψρAA′)

∥∥σAB)

where ρA = UA(g)ρAU†A(g), i.e. ρA is invariant under G.

• Covariance group of N⊗n always contains permutations Sn.

• Thus, we can restrict the optimization in Rα(N⊗n) topermutation invariant states ρAn .

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Step 4: Asymptotic Sub-Additivity

• Employ the fact that ψρAnA′n is in the symmetric subspace:

ψρAA′ ≤ PsymmAnRn ≤ n|A|

2

∫dµ(θ) θ⊗nAR .

• The quantum way to restrict to product states in the converse.

• This allows us to show (skipping a few technical steps) that

Rα(N⊗n) ≤ nRα(N ) +O(log(n)).

• Hence, Qε(N ) ≤ Rα(N ) for all α > 1.

• And, thus, by continuity as α→ 1, we find Qε(N ) ≤ R(N ).

• A more detailed analysis reveals that the fidelity convergesexponentially fast to 0 for any d > R(N ).

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Result 2: Dephasing Channels Satisfy Strong Converse

• For all quantum channels we thus have

Ic(N ) ≤ Q(N ) ≤ Qε(N ) ≤ R(N )

for all ε ∈ (0, 1).

Result

For generalized dephasing channels Z, we have Ic(Z) = R(Z).

• The inequalities collapse and Qε(Z) = Q(Z).

• Includes qubit dephasing channel:

Zλ : ρ 7→ (1− λ)ρ+ λZ ρZ .(a cc† b

)7→(

a (1− 2λ)c(1− 2λ)c† b

)

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Summary and Outlook

• Our strong converse also holds with one-way classicalcommunication assistance. However, it is unclear if it alsoholds for two-way assistance.

• For more details on this result, see:

arX

iv:1

406.

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v2 [

quan

t-ph]

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Nov

201

4

1

Strong Converse Rates for Quantum CommunicationMarco Tomamichel∗, Mark M. Wilde†, Andreas Winter‡

Abstract

We revisit a fundamental open problem in quantum information theory, namely whether it is possible totransmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probabilityof decoding error. Here we establish that the Rains information of any quantum channel is a strong converse ratefor quantum communication: For any code with a rate exceeding the Rains information of the channel, we showthat the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if weconsider codes that perform classical post-processing on the transmitted quantum data.

Our result has several applications. Most importantly, for generalized dephasing channels we show that theRains information is also achievable, and thereby establish the strong converse property for quantum communicationover such channels. This for the first time conclusively settles the strong converse question for a class of quantumchannels that have a non-trivial quantum capacity. Moreover, we show that the classical post-processing assistedquantum capacity of the quantum binary erasure channel satisfies the strong converse property.

I. INTRODUCTION

The quantum capacity of a quantum channel N , denoted Q(N ), is defined as the maximum rate (inqubits per channel use) at which it is possible to transmit quantum information over many memoryless usesof the channel, such that a receiver can recover the state with a fidelity that asymptotically converges toone as we increase the number of channel uses. (We will formally introduce capacity in Section II-B.) Thequestion of determining the quantum capacity was set out by Shor in his seminal paper on quantum errorcorrection [1]. Since then, a number of works established a “multi-letter” upper bound on the quantumchannel capacity in terms of the coherent information [2]–[4], and the coherent information lower boundon quantum capacity was demonstrated by a sequence of works [5]–[7] which are often said to bear“increasing standards of rigor.”1 In more detail, the work in [5]–[7] showed that the following coherentinformation of the channel N is an achievable rate for quantum communication:

Ic (N ) := maxφRA

I(R⟩B)ρ, where ρRB = NA→B (φRA) (1)

and the optimization is over all pure bipartite states φRA. Here, the coherent information of a bipartite stateρRB is defined as I(R⟩B)ρ := H(ρB)−H(ρRB), with the von Neumann entropies H(ρ) := −Tr{ρ log ρ}.2From the above result, we can also conclude that the rate Ic

!N ⊗l

"/l is achievable for any positive integer l,

simply by applying the formula in (1) to the “superchannel” N ⊗l and normalizing. By a limiting argument,we find that the regularized coherent information liml→∞ Ic

!N ⊗l

"/l is also achievable, and Refs. [2]–[4]

established that this regularized coherent information is also an upper bound on quantum capacity. Thisestablishes that

Q(N ) = liml→∞

Ic(N ⊗l)

l. (2)

Clearly, the regularized coherent information is not a tractable characterization of quantum capacity. Butthe later work of Devetak and Shor proved that the “single-letter” coherent information formula in (1) is

∗ School of Physics, The University of Sydney, Sydney, NSW 2006, Australia. (Email: [email protected]).† Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana

State University, Baton Rouge, Louisiana 70803, USA.‡ ICREA & Fısica Teorica, Informacio i Fenomens Quantics, Universitat Autonoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain.1However, see the later works [8] and [9], which set [5] and [6], respectively, on a firm foundation.2All logarithms in this paper are taken base two.

arXiv:1406.2

946

• The techniques also allow to establish a one-shot“meta-converse” in terms of binary quantum hypothesistesting (see arXiv: 1504.04617).

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