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On the quantum Renyi relative entropies and related capacity formulas

M. Mosonyi, F. Hiai

arXiv:0912.1286v4quant-ph

TL;DR

The paper addresses how quantum α-relative entropies and related channel capacities can be given operational meaning. It represents these entropies through generalized cutoff rates, proves equivalence of several Rényi-based capacities for α∈(0,2], and bounds one-shot transmission capacity.

  • Problem

    Operationally interpreting statistical-distance measures of correlations and obtaining finite-use capacity bounds remain central questions for quantum and classical-quantum channels.

  • Method

    The paper uses generalized cutoff rates for quantum state discrimination and minimax-based convexity arguments to analyze Rényi-relative-entropy capacities of classical-quantum channels.

  • Results

    For α∈(0,2], the various Rényi-based capacities coincide, and the paper gives an upper bound on one-shot ε-capacity under compact-image assumptions.

  • Takeaways & Limitations

    Quantum α-relative entropies receive an operational interpretation through cutoff rates, while the capacity equivalence identifies a common formulation across α∈(0,2].

Abstract

from arXiv · show

We show that the quantum $α$-relative entropies with parameter $α\in (0,1)$ can be represented as generalized cutoff rates in the sense of [I. Csiszar, IEEE Trans. Inf. Theory 41, 26-34, (1995)], which provides a direct operational interpretation to the quantum $α$-relative entropies. We also show that various generalizations of the Holevo capacity, defined in terms of the $α$-relative entropies, coincide for the parameter range $α\in (0,2]$, and show an upper bound on the one-shot epsilon-capacity of a classical-quantum channel in terms of these capacities.

I. INTRODUCTION

The introduction motivates operationally meaningful quantum statistical distances and develops Rényi-capacity generalizations for classical-quantum channels. It establishes capacity equivalence for α∈(0,2] and an upper bound for one-shot transmission.

  • Motivation: Statistical distances are useful state-distance measures that need not satisfy metric axioms but obey convexity and data-processing-type properties.Relative entropy is presented as the most popular example.
  • Motivation: Rényi relative entropies lacked a direct operational interpretation in the quantum setting, motivating generalized cutoff rates.Csiszár’s classical cutoff-rate construction supplies the relevant model.
  • Channel framework: A classical-quantum channel maps inputs x∈X to quantum states W_x, with a lifting that records the classical input alongside the output.The lifted state is δ_x⊗W_x, and averaging over p produces the joint input-output state.
  • Capacity generalizations: For relative entropy, the Holevo capacity is the maximal input-output correlation and equals the relative-entropy radius of the channel range.These equivalent formulations motivate several Rényi-based capacity generalizations.
  • Main contributions: The Rényi capacity definitions coincide for classical-quantum channels when α∈(0,2].This extends previously described equalities to the stated quantum-channel setting.
  • Main contributions: An upper bound on one-shot ε-capacity is obtained in terms of Rényi capacities with α>1, complementing earlier lower-bound results.The bound is described as asymptotically optimal in the relevant limit.

II. PRELIMINARIES ON THE RÉNYI RELATIVE ENTROPIES

This section defines Rényi relative and quasi-relative entropies and develops their support, monotonicity, convexity, and positivity properties. It also relates Rényi relative entropies to Hoeffding distances and discusses the max-relative entropy.

  • Definitions: The α-quasi-relative entropy is introduced for positive semidefinite operators, alongside conventions for supports, powers, logarithms, and density operators.These conventions support the subsequent operator inequalities.
  • Definitions: The Rényi relative entropy is defined through Tr A^αB^(1−α), with support conditions determining when its value is finite or infinite.For α∈(0,1), the expression is finite under the stated conventions; for α>1, support inclusion is required.
  • Structural properties: The underlying quasi-relative quantities are monotone under completely positive trace-preserving maps and jointly convex for α∈[0,2]\{1}.For α>1, joint convexity does not automatically transfer to the corresponding Rényi relative entropies.
  • Structural properties: For α∈[0,2], the Rényi relative entropy is convex in its second operator argument when the first operator is fixed.The proof for α∈(1,2] uses a positive linear functional and operator-monotone decreasing functions.
  • Structural properties: When Tr A≤1, S_α(A||B) increases with α separately on [0,1) and (1,+∞), with full-range monotonicity when Tr A=1.Under the stated trace conditions, the quantities also satisfy non-negativity and characterize equality in specified cases.
  • Hoeffding distances: Hoeffding distances and Rényi relative entropies mutually determine one another through a Legendre-Fenchel transform for α∈[0,1).The Hoeffding distance is monotone decreasing in its parameter and is bounded above by the relative entropy at parameter zero.
  • Related divergences: For noncommuting operators, max-relative entropy can differ from the α→∞ Rényi limit, while S_2≤S_max≤S_∞ generally holds.Equality between max-relative entropy and the limit is noted in the commuting case.

III. CUTOFF RATES FOR QUANTUM STATE DISCRIMINATION

The section develops cutoff rates for quantum state discrimination and shows that, for α∈(0,1), they provide an operational interpretation of Rényi relative entropies. It also extends the result to correlated state sequences under regularity assumptions.

  • The analysis minimizes the second-kind error probability subject to an exponential constraint on the first-kind error probability.
  • The Hoeffding distance gives the optimal exponential decay rate, but evaluating it generally requires all Rényi relative entropies and an optimization.
  • Cutoff rates provide linear approximations to the Hoeffding-distance function that are easy to evaluate and guarantee a decay rate for r below the cutoff.
  • For every α∈(0,1), the resulting cutoff-rate characterization gives an operational interpretation of the quantum Rényi relative entropies.
  • The framework extends from independent identically distributed trials to correlated state sequences when the scaled Rényi quantities converge uniformly and the relevant function is differentiable.
  • The stated assumptions include examples from finite-state classical ergodic Markov chains and finite-block restrictions of non-interacting fermionic and bosonic temperature states.

IV. EQUIVALENCE OF CAPACITIES

For compact-image classical-quantum channels, the paper establishes equality among several Rényi-relative-entropy capacity formulations for α∈(0,2]. The proof uses compactness, continuity, convexity, and minimax arguments, extending the equivalence to both Sα and Qα formulations.

  • Setup: The channel range is modeled as a compact subset K of the state space, with probability measures M(K) and finitely supported measures Mf(K).This compact framework supports the subsequent continuity and minimax arguments.
  • Analytic ingredients: The auxiliary functions fα,ε and gα,ε are affine and continuous in the measure argument, and convex and continuous in the state argument for ε>0.These properties provide the structural conditions needed for the optimization steps.
  • Optimization: For every ε>0, an optimizing state σε exists, and the corresponding minimax identities remain valid when maxima over all probability measures are restricted to finitely supported measures.The existence follows from compactness and semicontinuity, while the equality uses Sion’s minimax theorem.
  • Extension: The same existence and equality relations hold for gα,ε with Qα replacing fα,ε with Sα.This transfers the optimization framework from Rényi relative entropy to the Qα formulation.
  • Capacity equivalence: For α∈(0,2], the capacities defined in (8)–(10) are equal for classical-quantum channels with compact image.The result includes α=1 and assumes compactness of ran W.

V. THE ONE-SHOT CLASSICAL CAPACITY OF QUANTUM

The section develops one-shot and asymptotic capacity bounds for classical-quantum channels using Rényi-relative-entropy and related quantities. The upper bound is asymptotically sharp, yielding the Holevo capacity in the limit.

  • Coding framework: The channel model uses codes with message sets, encoding maps, and positive-operator-valued decoding measurements, together with i.i.d. channel extensions for blocklength n.The rate is defined from the normalized logarithm of the message-set size.
  • One-shot bounds: The one-shot ε-capacity is bounded using generalized Holevo capacities defined through Hoeffding distances and Rényi relative entropies.The construction relates one-shot message transmission to Sα,0(W) and the corresponding Hoeffding-capacity quantity.
  • Asymptotic behavior: The one-shot bound may be loose for a single channel use but is asymptotically optimal because it yields the Holevo capacity as a lower bound on the optimal asymptotic rate.This establishes the operational relevance of the bound despite its possible finite-use looseness.
  • Asymptotic behavior: The upper bound becomes asymptotically sharp and gives the Holevo capacity as an upper bound on optimal information-carrying capacity.The result applies to code sequences with rates and error criteria as specified in the section.
  • Upper bound: For compact ran W, Theorem V.2 provides an upper bound on the one-shot ε-capacity for every ε∈[0,1).The theorem is derived from success-probability bounds for arbitrary codes.

VI. REMARKS ON THE DIVERGENCE RADIUS

The section relates divergence radii to channel coding and state discrimination, while examining geometric properties of their centers. Unlike the relative-entropy center, a general Rényi center need not be unique or lie in the closed convex hull.

  • Operational roles: Rényi divergence radii of channel ranges relate to direct coding for α∈[0,1) and converse coding for α∈(1,+∞], with α near 1 asymptotically relevant.For state discrimination, the relevant quantities are connected to divergence radii as well.
  • State discrimination: For two states, the optimal Helström family has parameter q=exp(−RSmax({ρk})), and its mixed states form an Smax-center.For r=2, the optimal success probability is Ps=(1+D)/2 with D=(1/2)||ρ1−ρ2||1.
  • Continuity bounds: The section states a continuity bound for bounded functions satisfying the specified binary-mixture condition, expressed through trace distance and binary entropy.The resulting bound applies to functions on pairs of quantum states.
  • Applications: The same condition applies to von Neumann entropy, conditional entropy, and relative-entropy distance from a convex set containing a faithful state.For the latter quantities, the stated bound slightly improves earlier results.
  • Geometric centers: The relative-entropy center is unique and lies inside the closed convex hull, but this property generally fails for other Rényi relative entropies.The section gives an S∞ example whose center does not lie on the line segment joining two states.

VII. CONCLUDING REMARKS

The paper interprets quantum Rényi relative entropies through generalized cutoff rates and establishes capacity identities and one-shot bounds, while identifying limitations for converse exponents and finite-size capacities.

  • Operational interpretation: The analysis represents Rényi relative entropies as generalized cutoff rates using a quantum-compatible definition of Hoeffding distances.The approach avoids the classical Hellinger-arc argument and relies on a definition whose operational meaning follows from the quantum Hoeffding bound.
  • Capacity identities: Minimax arguments establish identities among Rényi-capacity definitions, with Theorem II.1 supplying a new quantum result.The proof requires convexity properties that fail to extend beyond α=2 in the quantum setting.
  • Capacity regimes: The α∈(0,1) and α∈(1,+∞) regimes correspond respectively to direct and strong-converse information-theoretic tasks.Below capacity, error probabilities decay exponentially; above capacity, success probabilities decay exponentially.
  • Limitations: Exact quantum converse error exponents are unknown, preventing extension of the classical cutoff-rate results for κ>0.Known results provide inequalities between cutoff rates and Rényi relative entropies that are expected to become equalities.
  • One-shot bounds: Finite-size effects make one-shot capacities discontinuous in ε and more dependent on channel parameters than asymptotic capacities.The paper therefore frames universal one-shot results as lower and upper estimates rather than similarly compact formulas.
  • One-shot bounds: The one-shot capacity bounds are not shown to be optimal, although applying them to multiple channel copies recovers the optimal asymptotic capacity.This recovery occurs as the number of copies tends to infinity.

APPENDIX A A MINIMAX THEOREM

Appendix A states a minimax theorem under compactness, lower semicontinuity, and finite-subset conditions, then derives a monotonicity-based corollary.

  • A. Minimax theorem: Minimax theorems provide conditions under which the inequality between infimum-supremum and supremum-infimum expressions becomes equality.The appendix introduces Lemma A.1 as a step toward Sion’s minimax theorem.
  • A. Minimax theorem: Under compactness of X, lower semicontinuity of f(·,y), and the finite-subset condition, the infima can be replaced by minima.Lower semicontinuity makes relevant level sets closed and compact, enabling finite subcover arguments.
  • A. Minimax theorem: The proof selects finitely many y-values whose lower-level sets have empty intersection, yielding the reverse minimax inequality.The argument applies for every c below the relevant infimum-supremum value.
  • A. Corollary: If f(x,·) is monotonic, the infima in the minimax expressions can be replaced with minima under the appendix’s compactness and lower-semicontinuity assumptions.The monotonicity condition reduces any finite subset of Y to an extremal element.

APPENDIX B THE LIMIT OF THE α-CAPACITIES

Appendix B establishes continuity and monotonicity properties of α-capacity functions and uses them with minimax arguments to analyze their limit near α=1.

  • B. Assumptions: The appendix’s results rely on a fixed classical-quantum channel W and compactness of its range where stated.The compact-range assumption is explicit in the continuity lemma and related propositions.
  • B. Capacity properties: For α>1, the function p↦χ_α(p) is concave on the set of finitely supported probability measures.The proof derives concavity from the defining inequality and the concavity of the relevant logarithmic expression.
  • B. Capacity properties: The α-capacity functions χ_α(p) and χ̃_α(p) are monotonically increasing in α.This follows from monotonicity of the Rényi relative entropies.
  • B. Limits at α=1: lim α→1 χ_α(p)=χ_1(p) and lim α→1 χ̃_α(p)=χ̃_1(p).The proof treats the limits from α above and below one separately.
  • B. Compactness and continuity: When ran W is compact, a topology can be placed on finitely supported measures so that the space is compact and χ̃_α is continuous.The construction uses a compact probability simplex-product representation and a factor topology.
  • B. Minimax application: The appendix applies minimax to f(p,α):=−χ̃_α(p), using continuity in p and monotonic decrease in α.These properties supply the conditions needed for the minimax corollary.
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