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Strong converse exponent for classical-quantum channel coding

Milan Mosonyi, Tomohiro Ogawa

arXiv:1409.3562v8quant-phcs.ITmath-phmath.ITmath.MP

TL;DR

Classical-quantum channel coding calls for an exact strong-converse exponent, while quantum Rényi divergences admit inequivalent definitions. The paper gives the exponent above the Holevo capacity as an Arimoto-type transform of Rényi capacities for α>1 based on D* divergences.

  • Problem

    Quantum states admit inequivalent Rényi divergences, and thus channel Rényi capacities, leaving the appropriate capacity family for strong-converse exponent expressions at issue.

    Source: §I · p. 2

  • Method

    The paper derives the classical-quantum strong-converse exponent as an Arimoto-type transform of Rényi α-capacities for α>1 based on D* divergences.

    Source: §I · p. 2

  • Results

    The paper determines the exact strong-converse exponent for classical-quantum channels at every rate above the Holevo capacity.

    Source: §I · p. 2, A. Classical-quantum channel coding… · p. 27

  • Takeaways & Limitations

    The expression gives an operational interpretation to capacities derived from D* divergences for strong-converse problems.

    Source: §I · p. 2

Abstract

from arXiv · show

We determine the exact strong converse exponent of classical-quantum channel coding, for every rate above the Holevo capacity. Our form of the exponent is an exact analogue of Arimoto's, given as a transform of the Renyi capacities with parameters alpha>1. It is important to note that, unlike in the classical case, there are many inequivalent ways to define the Renyi divergence of states, and hence the R\'enyi capacities of channels. Our exponent is in terms of the Renyi capacities corresponding to a version of the Renyi divergences that has been introduced recently in [M\"uller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013)], and [Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014)]. Our result adds to the growing body of evidence that this new version is the natural definition for the purposes of strong converse problems.

I. INTRODUCTION … C. Minimax theorems

The paper determines the exact strong converse exponent for classical-quantum channel coding at rates above the Holevo capacity. It expresses the exponent through Rényi capacities based on D*α divergences and develops the mathematical tools needed to establish that result.

I. INTRODUCTION

  • For rates above the Holevo capacity, the exact strong converse exponent is an Arimoto-type transform of the channel’s Rényi capacities for α > 1.This gives the exponent as a function of coding rate.Source: §I · p. 2
  • The exponent uses D*α divergences, introduced in [50] and [75], and gives their channel capacities an operational interpretation for classical-quantum coding.The paper presents D*α as a natural trade-off quantifier in converse problems, while distinguishing it from the Dα family associated with direct domains.Source: §I · p. 2
  • Block pinching converts a suboptimal D♭α-capacity expression into the correct exponent, following a relative-entropy optimization bound for classical-quantum channels.The paper connects mutual informations of product channels and their pinched versions to carry out this conversion.Source: §I · pp. 2–3
  • The exponent formula also applies to entanglement-breaking channels and group-covariant channels with additive minimum output α-entropy.Source: §I · p. 3

A. Notations and basic lemmas

  • The preliminaries define operator powers on an operator’s support and represent finite-outcome measurements as positive trace-preserving maps with commutative outputs.Source: A. Notations and basic lemmas · pp. 3–4

B. Convexity

  • The convexity preliminaries give conditions for pointwise suprema and affine reparameterizations to preserve convexity.They also define joint convexity for functions on products of convex sets.Source: B. Convexity · p. 4

C. Minimax theorems

  • A minimax lemma from [47] gives the stated optimization equality when the function is lower semicontinuous in a compact variable and monotone in the ordered variable.Under these conditions, the infima can be replaced by minima.Source: C. Minimax theorems · p. 5
  • The Kneser–Fan and Sion [42, 69] minimax conditions yield the stated optimization equality for compact convex X and convex Y.The lemma permits concave or quasi-concave, upper-semicontinuous dependence on Y, alongside convex or quasi-convex, lower-semicontinuous dependence on X.Source: C. Minimax theorems · p. 5

D. Universal symmetric states … A. Definitions and basic properties

This section introduces universal symmetric states and classical-quantum channel notation, and develops basic properties and variational representations of Rényi divergences. It also treats support dependence, relations between variational expressions, uniqueness and convexity, and asymptotic measurement characterization.

D. Universal symmetric states

  • For each finite-dimensional H and n, a universal symmetric state exists, dominates every symmetric state, and has at most v_{n,d} distinct eigenvalues.A construction is reviewed from [27]; [29, Lemma 1] gives a different existence argument.Source: D. Universal symmetric states · p. 6

E. Classical-quantum channels

  • A classical-quantum channel maps inputs from an arbitrary alphabet to quantum states on a finite-dimensional output space, and its n-fold extension acts on n-tuples with n-fold output states.Source: E. Classical-quantum channels · p. 6

A. Definitions and basic properties

  • For noncommuting states, the considered Rényi divergence definitions can differ, while all three coincide with the classical expression for commuting states.The passage identifies one family as quantum f-divergences and notes differing operational roles for the divergence families.Source: A. Definitions and basic properties · p. 7
  • For positive semidefinite operators, the regularized divergence limits exist and agree, recover the invertible-operator formulas, and are infinite for α>1 unless supp ρ ⊆ supp σ.The support condition also determines when the corresponding Q quantities are infinite.Source: A. Definitions and basic properties · pp. 7–9
  • The stated ψ^t_α continuity result gives continuity for all α>0 when supp ρ ⊆ supp σ, and continuity on (0,1) with a jump at 1 otherwise.Source: A. Definitions and basic properties · p. 8
  • The ♭ quantity has maximum-over-positive-operators and negative minimum-over-states variational representations; when D♭α(ρ∥σ) is finite and α∈(0,+∞)\{1}, the optima in (38)–(39) are unique.The representations (37)–(38) also hold at α=1.Source: A. Definitions and basic properties · p. 10
  • Strict positivity of relative entropy establishes that τα is the unique minimizer.The passage states that this follows from identity (46), whose expression is not included here.Source: A. Definitions and basic properties · p. 12
  • Theorem III.6 connects variational expressions (37) and (38), extends them to non-invertible states, and shows that either expression yields the other.The proof obtains (37) from Tropp’s argument [73] or proves (38) first and derives (37).Source: A. Definitions and basic properties · p. 12
  • D*α can be recovered asymptotically from Rényi divergences of commuting operators and as the largest post-measurement Rényi divergence over many copies.The passage attributes the first identity to [48] and its extension to [29], and derives the second using monotonicity under measurements [21].Source: A. Definitions and basic properties · p. 12
  • The convexity arguments establish convexity in α for ψ*_α and the corresponding Q quantities using limits and variational representations.The proof derives convexity of ψ*_α as a limit of convex functions and uses a supremum-of-convex-functions argument for the corresponding quantities.Source: A. Definitions and basic properties · p. 13

B. Convexity and monotonicity

The section establishes maximal ranges of joint convexity for three quantum Rényi divergences and gives partial convexity results in the second argument. It also shows that Q♭_α is not monotone under CPTP maps for any α>1.

B. Convexity and monotonicity

  • Joint convexity holds exactly for α∈[0,2] when t={ }, α∈[1/2,+∞) when t=*, and α∈[0,1] when t=♭.These are the maximal intervals for the respective divergence versions.Source: B. Convexity and monotonicity · p. 14
  • For α>1, Q♭_α fails joint convexity.The section constructs positive solutions that provide counterexamples.Source: B. Convexity and monotonicity · p. 15
  • For every α>1, Q♭_α is not monotone under CPTP maps, and it is not even monotone under pinching by the reference operator.Source: B. Convexity and monotonicity · p. 15
  • With ρ fixed, D^t_α(ρ∥·) is convex for t=♭ when α∈[0,+∞), t=* when α∈[1/2,+∞), and t={ } when α∈[0,2].The maps σ↦D♭_∞(ρ∥σ) and σ↦D*_∞(ρ∥σ) are also convex.Source: B. Convexity and monotonicity · pp. 15–16

C. Further properties and relations

The section establishes structural properties of the Rényi divergences: tensor-product additivity, an ordering among divergence variants, and a positivity bound with a precise equality condition. It also gives regularization and semicontinuity properties used in later arguments.

C. Further properties and relations

  • The Q quantities are multiplicative, so the corresponding Rényi divergences are additive on tensor powers.The stated additivity holds for every n∈N.Source: C. Further properties and relations · p. 16
  • The divergences satisfy an ordering: D*α(ρ∥σ)≤Dα(ρ∥σ), while Dα(ρ∥σ)≤D♭α(ρ∥σ) for α∈(0,1) and Dα(ρ∥σ)≥D♭α(ρ∥σ) for α>1.The proof relates the first inequality to the Araki–Lieb–Thirring inequality and the latter inequalities to Golden–Thompson.Source: C. Further properties and relations · p. 16
  • For α>0 and all three divergence variants, the divergence obeys a lower bound with equality exactly when ρ is a constant multiple of σ.For states, this establishes strict positivity except when the states coincide.Source: C. Further properties and relations · p. 17
  • The regularization lemma gives monotonicity in ε for the stated Q quantities and a two-parameter infimum–supremum relation, also valid for D♭ when α∈(0,1).The lemma covers each of the three divergence variants.Source: C. Further properties and relations · pp. 17–18
  • For α in the applicable range A(t), the divergences are lower semicontinuous in σ; they are also upper semicontinuous under the stated conditions, including for D♭ when α∈(0,1).The lower-semicontinuity claim also covers ψ∞; the passage specifies upper semicontinuity for the flat variant in the indicated range.Source: C. Further properties and relations · p. 19

IV. R´ENYI CAPACITIES

The section motivates Rényi generalizations of the Holevo capacity as a refinement for describing the coding-rate and decoding-error trade-off, and gathers definitions needed for the paper’s strong converse exponent proof.

IV. R´ENYI CAPACITIES

  • The Holevo-Schumacher-Westmoreland theorem [37, 65] identifies the asymptotic classical information transmission capacity of a quantum channel with its Holevo capacity under vanishing error.The passage refers readers to Section V A for details.Source: §IV · p. 19
  • The paper expects Rényi generalizations of the Holevo capacity to refine the trade-off between coding rate and decoding error.It states that Theorem V.2 shows this is the case for the strong converse exponent of classical-quantum channels.Source: §IV · p. 19
  • This section collects definitions and technicalities needed later in the proof of the main result.Source: §IV · p. 19

A. R´enyi mutual informations and equivalent definitions of the R´enyi capacities

The section defines two Rényi analogues of the Holevo quantity and shows that, after optimization over input distributions, their differences disappear for specified divergence types and parameter ranges. This equivalence lets the authors define channel Rényi capacities consistently and switch between forms used in the strong converse proof.

A. R´enyi mutual informations and equivalent definitions of the R´enyi capacities

  • Replacing relative entropy with a Rényi divergence can make the equivalent Holevo formulas differ, so the section defines two Rényi-Holevo quantities.The two definitions are exact analogues of the second formulas for the ordinary Holevo quantity.Source: A. R´enyi mutual informations and… · pp. 19–20
  • After optimization, the Rényi-Holevo expressions coincide for t = { } when α ∈ (0, 2], t = * when α ∈ [1/2, +∞), and t = ♭ when α ∈ (1, +∞).For t = ♭, two of the expressions also coincide when α ∈ (0, 1).Source: A. R´enyi mutual informations and… · p. 20
  • The equivalence proof uses minimax arguments with compactness of the state space and an intermediate regularization to handle infinite Rényi divergences.The input distributions need not form a compact set, and divergences can be infinite when the reference state is not invertible.Source: A. R´enyi mutual informations and… · p. 22

B. R´enyi capacities of pinched channels

This section develops the relation between Rényi capacity quantities for a channel and its pinched tensor-power channel, as a technical tool for determining the strong converse exponent. It also establishes additivity and symmetry properties used in that analysis.

B. R´enyi capacities of pinched channels

V. THE STRONG CONVERSE EXPONENT FOR CLASSICAL-QUANTUM CHANNELS · A. Classical-quantum channel coding and the strong converse exponent

The paper gives an exact Arimoto-style expression for the strong converse exponent of classical-quantum channel coding as a transform of Rényi capacities with parameters α>1. The converse Hoeffding capacity based on D*α is operationally relevant, while the one based on D♭α also plays a role in the proof.

A. Classical-quantum channel coding and the strong converse exponent

  • The channel capacity equals the Holevo capacity χ(W), by the Holevo-Schumacher-Westmoreland theorem [37, 65].Rates below capacity admit codes with asymptotically vanishing error probability.Source: A. Classical-quantum channel coding… · p. 26
  • Above the Holevo capacity, every code sequence has error probability tending to 1 exponentially, and its success probability decays exponentially.The optimal decay exponent of the success probability at rate R is the strong converse exponent.Source: A. Classical-quantum channel coding… · p. 26
  • The strong converse exponent has an exact analogue of the Arimoto-Dueck-Körner exponent for classical channels.Theorem V.2 gives the expression for any rate R ≥ 0.Source: A. Classical-quantum channel coding… · p. 27
  • The theorem’s novel contribution is the converse inequality sc(R, W) ≤ H∗R,c(W), while its optimality part follows by standard techniques.The authors identify this converse inequality as the theorem’s achievability part.Source: A. Classical-quantum channel coding… · p. 27
  • The exponent expression directly parallels the Hoeffding anti-divergence used for binary state discrimination [48].The corresponding expression extends to other Rényi capacities as converse Hoeffding capacities.Source: A. Classical-quantum channel coding… · p. 27
  • The converse Hoeffding capacity corresponding to D*α is operationally relevant, while the D♭α counterpart also plays an important role in the proof.Source: A. Classical-quantum channel coding… · p. 27

B. Lower bound for the strong converse exponent · C. Dueck-K¨orner exponent

Section B states a lower-bound lemma for the strong converse exponent using the new Rényi relative entropies. Section C develops a Dueck–Körner-type upper bound and alternative representations of its variational exponent for classical–quantum channels.

B. Lower bound for the strong converse exponent

C. Dueck-K¨orner exponent

  • For classical channels the bounds in (109) and (111) coincide, whereas for quantum channels they need not; block pinching later yields an upper bound matching (109).The two bounds together give Theorem V.2 in the classical case.Source: C. Dueck-K¨orner exponent · p. 28
  • For any rate R > 0 and input distribution P, the Dueck–Körner bound gives sc(R, W) ≤ F(P, R, W), where F minimizes D(V ∥ W|P) + |R − χ(V, P)|+ over channels V.The exponent F is defined as the infimum of this expression over classical–quantum channels V.Source: C. Dueck-K¨orner exponent · pp. 28–29
  • The variational exponent splits into F1 over channels with χ(V, P) > R and F2 over channels with χ(V, P) ≤ R.F1 minimizes D(V ∥ W|P), while F2 minimizes D(V ∥ W|P) + R − χ(V, P); separate lemmas bound sc(R, W) by each term.Source: C. Dueck-K¨orner exponent · pp. 29–30
  • The F1 bound uses a dummy-channel lemma and random coding with the Hayashi–Nagaoka decoder to show rates above F1 are R-achievable.The proof draws codewords independently from P^⊗n and uses the dummy channel to control success rates.Source: C. Dueck-K¨orner exponent · pp. 29–30
  • The F2 bound constructs an R-rate code by expanding a code achievable at a lower rate, yielding sc(R, W) ≤ F2(P, R, W).The construction uses a channel V with χ(V, P) ≤ R and a lower-rate code supplied by the F1 bound.Source: C. Dueck-K¨orner exponent · pp. 30–31
  • Given W ∈ C(H|X) and R ≥ 0, for any P ∈ Pf(X), F(P, R, W) has the representation min V ∈ CW,P {D(V ∥ W|P) + |R − χ(V, P)|+}.Source: C. Dueck-K¨orner exponent · p. 31

D. Strong converse exponent: achievability

This section proves the achievability part of the strong converse exponent result. Theorem V.14 states a bound for every classical-quantum channel and every R > 0.

D. Strong converse exponent: achievability

  • Theorem V.14 gives the achievability bound for any classical-quantum channel and any R > 0.The supplied passage omits the displayed formula, so the bound's explicit form is unavailable here.Source: D. Strong converse exponent:… · p. 34
  • The proof uses a universal symmetric state σu,m to define pinching and takes decoding POVM elements invariant under tensor-power pinching, preserving success probability.The passage also invokes Theorem V.6 and equation (134) to obtain a sequence of codes C(m).Source: D. Strong converse exponent:… · pp. 34–35
  • For each n, the proof chooses k with km ≤ n < (k + 1)m and constructs a code Cn for W⊗n from C(m).The decoder is defined using decoding POVM elements from C(m).Source: D. Strong converse exponent:… · p. 35

VI. APPLICATION TO QUANTUM CHANNELS

The section applies the coding framework to quantum channels, where codewords may be entangled across channel uses. This differs from the classical–quantum setting, whose product extension allows only product inputs.

VI. APPLICATION TO QUANTUM CHANNELS

  • A quantum channel is a completely positive trace-preserving linear map between operator spaces of finite-dimensional input and output Hilbert spaces.Source: §VI · p. 36
  • Quantum-channel codewords may be arbitrary density operators on H_in^⊗n, so they can be entangled across channel uses.The n-fold extension of a quantum channel can take entangled inputs, unlike its classical–quantum extension, which can take only product inputs.Source: §VI · p. 36

A. Entanglement breaking channels · B. Covariant channels

For entanglement-breaking channels, the paper extends the relevant subadditivity property to every α>1 and identifies the strong converse exponent via (149). For KW-class channels, it develops covariance-based results and states a strong converse exponent theorem.

A. Entanglement breaking channels

  • The subadditivity property for entanglement-breaking channels holds for every α>1, yielding their strong converse exponent in (149).Earlier work established the property only for α∈(1,2]; Propositions III.18 and IV.2 extend the range.Source: A. Entanglement breaking channels · p. 36

B. Covariant channels

  • The KW-class consists of group-covariant channels with additive minimum output α-entropy for all α>1, following [41].Group covariance requires an irreducible output representation.Source: B. Covariant channels · pp. 36–37
  • Depolarizing channels [39] and unital qubit channels are examples of the KW-class.Source: B. Covariant channels · p. 37
  • Lemma VI.2 establishes a Rényi-quantity bound for group-covariant channels throughout α∈[1/2,+∞).Its proof uses output-representation irreducibility and Proposition III.18.Source: B. Covariant channels · pp. 37–38
  • The cited passages give α(Φ) for n∈N and α∈[1/2,+∞), and state that Lemma VI.2 and (150) yield the following result.Source: B. Covariant channels · p. 38
  • Theorem VI.4 gives a strong converse exponent result for every rate R>0 for channels in the KW-class.The theorem follows Corollary VI.3 and Theorem VI.1.Source: B. Covariant channels · p. 38

Appendix A: Universal symmetric states · Appendix B: The auxiliary function

Appendix A reviews a universal symmetric-state construction, while Appendix B establishes concavity results for auxiliary functions under specified assumptions and for a particular case with α>1.

Appendix A: Universal symmetric states

  • Appendix A reviews the universal symmetric-state construction [27], using Schur–Weyl duality [23] to represent permutation-invariant states blockwise and establish that σu,n satisfies Lemma II.8’s criteria.In the decomposition, each block is described by a density operator and a probability weight on the relevant Young diagrams.Source: Appendix A · pp. 38–39

Appendix B: The auxiliary function

  • For classical-quantum W, the defined quantities are generalized Holevo quantities; the auxiliary-function formulas use maxima for 0<α<1 and minima for α>1.The latter ranges correspond respectively to s>0 and −1<s<0.Source: Appendix B · pp. 39–40
  • If α ↦ ψ^(t)_α(ρ∥σ) is convex on (0,1), both auxiliary functions E^(t)_{0,i}(s,W,P) are concave for s>0 and i=1,2.This is Proposition B.1’s condition and conclusion for any map W and finitely supported probability distribution P.Source: Appendix B · p. 40
  • The quantities for Q^(t)_α with t={ }, *, and ♭ satisfy Proposition B.1’s convexity assumption, so their corresponding auxiliary functions are concave.This is Corollary B.2’s application of the general result.Source: Appendix B · p. 40
  • For t={ }, the Sibson identity [41] yields an explicit expression for E^(t)_{0,1}(s,W,P).The passage identifies this as a special case.Source: Appendix B · p. 40
  • The authors say their proof is simpler than the recent approach in and extends to auxiliary functions from other Rényi divergences and to i=2.These extensions are attributed to Proposition B.1 and Corollary B.2.Source: Appendix B · p. 40
  • For α>1 the Proposition B.1 proof fails because its infima need not preserve convexity, but E^♭_{0,i}(s,W,P) is concave on −1<s<0 for i=1,2.The limitation and the ♭-case result are stated separately in the appendix.Source: Appendix B · p. 40

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