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Quantum f-divergences and error correction

F. Hiai, M. Mosonyi, D. Petz, C. Beny

arXiv:1008.2529v6math-phcs.ITquant-ph

TL;DR

The paper asks how broadly quantum distinguishability measures retain monotonicity and equality-based reversibility under quantum maps, and how these properties support error correction. It develops a quantum f-divergence framework using operator convex functions, extends monotonicity and reversibility results, and applies them to error correction and related measures. The paper also supplies an integral representation for operator convex functions and identifies special cases where ordinary convexity suffices.

  • Problem

    The paper studies monotonicity and equality conditions for quantum f-divergences and related distinguishability measures, including their relevance to reversibility and quantum error correction.

  • Method

    It analyzes quantum f-divergences through operator convex functions, dual Schwarz maps, equality conditions, and an integral representation on the positive half-line.

  • Results

    The paper proves monotonicity under the dual of Schwarz maps for operator convex defining functions and extends reversibility results to a large class of f-divergences and distinguishability measures.

  • Takeaways & Limitations

    Preservation of suitable pairwise f-divergences by a stochastic map can provide a canonical stochastic reversal on a set of states, supporting quantum error correction.

  • Takeaways & Limitations

    The general analysis requires operator convexity, and whether this condition is necessary remains open; some special cases need only ordinary convexity.

Abstract

from arXiv · show

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well known distinguishability measures of quantum states are given by, or derived from, f-divergences; special examples include the quantum relative entropy, the Renyi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under the dual of Schwarz maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Holder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

1 Introduction

The paper develops quantum f-divergences as a framework for quantum distinguishability, focusing on their monotonicity under maps and the equality conditions that characterize reversibility. It extends these results to error correction and related measures while identifying operator convexity as central to the general theory.

  • Relative entropy measures quantum-state distinguishability, is operationally tied to Stein’s lemma, and underlies several information-theoretic quantities.
  • Quantum quasi-entropies generalize classical f-divergences, with relative and Rényi relative entropies arising from specific representing functions.
  • The paper proves monotonicity of f-divergences for operator convex functions under a common generalization of previously known map classes.
  • Equality in the monotonicity inequality is extended from relative entropy and selected Rényi entropies to a large class of f-divergences and implies reversibility under stated conditions.
  • The reversibility analysis is applied to quantum error correction and to preservation of Chernoff and Hoeffding distinguishability measures.
  • Operator convexity is essential to the general monotonicity and equality analysis, although whether it is necessary remains open; convexity suffices in special cases.

2 Quantum f-divergences: definition and basic properties

This section defines quantum f-divergences for positive semidefinite operators, including noninvertible reference operators, and establishes their basic representation, continuity, invariance, and uniqueness properties. It also relates these quantities to Rényi divergences and shows that quantum fidelity is not generally an f-divergence.

  • Quantum f-divergences are defined for positive semidefinite operators using a function f with suitable continuity and limiting behavior.
  • The defining limit exists and yields an explicit formula containing a support-dependent correction term when the second operator is noninvertible.
  • The divergence is represented through the relative modular operator L_A R_B^-1, with generalized inverses and support conventions handling singular operators.
  • For f_α(x)=x^α, the construction gives Tr A^αB^(1−α) under the stated support condition, yielding the corresponding Rényi relative entropy.
  • The representing function of an f-divergence is unique, and agreement of f-divergences on classical distributions implies agreement on quantum states.
  • Quantum fidelity cannot generally be represented as an f-divergence because its classical representing function produces Tr ρ^1/2σ^1/2 instead.
  • When ω(f) is finite, the f-divergence is continuous in its second variable, whereas finiteness is essential and continuity in the first variable requires an additional condition at zero.

3 Preliminaries on positive maps

This section establishes the map-theoretic framework for quantum f-divergence monotonicity, distinguishing positive, Schwarz, stochastic, and substochastic maps and developing support and trace-preservation tools. It also gives examples showing that Schwarz maps need not be 2-positive.

  • A linear map is positive, n-positive, or completely positive exactly when its Hilbert–Schmidt adjoint has the corresponding property; trace preservation corresponds to unitality of the adjoint.
  • The support projection B0 defines a corner algebra B0A1B0, and positivity preserves equality of supports between positive inputs.
  • Trace preservation on the support corner associated with B is equivalent to Tr Φ(B)=Tr B and to fixed-point conditions involving Φ∗(Φ(B)0).
  • For a trace-nonincreasing positive map, preservation of the traces of A and B is equivalent to trace preservation on the support of A+B.
  • The section derives a key positive-map inequality that yields monotonicity of Rényi 2-relative entropies and supports the later treatment of general f-divergences.
  • Schwarz maps are closed under composition, adjoints, and positive linear combinations, while Schwarz contractions are contractions and their adjoints define substochastic maps.
  • The family Φε contains parameter values that are Schwarz contractions but not 2-positive, showing that Schwarzness is strictly weaker than 2-positivity.

4 Monotonicity

This section proves monotonicity of quantum f-divergences under substochastic maps when the defining function is operator convex, then derives consequences for standard divergences, joint convexity, and broader map classes. It also identifies operator convexity as an essential proof assumption whose necessity remains unresolved.

  • 4.3 Monotonicity theorem: Theorem 4.3 establishes Sf(Φ(A)∥Φ(B)) ≤ Sf(A∥B) for operator convex f and substochastic Φ satisfying Tr Φ(B)=Tr B, under the theorem’s stated conditions.
  • 4.3 Monotonicity theorem: The proof reduces general operator convex functions to integral representations and primitive functions ϕt(x)=−x/(x+t), while separately controlling support and trace terms.
  • Applications: For 0<α<1, −Tr Φ(A)^αΦ(B)^(1−α) ≤ −Tr A^αB^(1−α), yielding monotonicity for the corresponding Rényi quantities.
  • Applications: The theorem yields monotonicity of Rényi relative entropies for α∈[0,2]\{1} and of the relative entropy generated by f(x)=x log x.
  • Limitations: The proof requires operator convexity, but whether this condition is necessary remains open; ordinary convexity suffices only in special cases discussed in Appendix A.
  • Further consequences: The same theorem implies joint convexity of f-divergences, making the class of maps preserving monotonicity convex and closed under composition.
  • Further consequences: Monotonicity also extends to co-substochastic maps because transposition leaves every f-divergence invariant, and to convex combinations of stochastic and co-stochastic maps.

5 Equality in the monotonicity

The paper characterizes when equality in f-divergence monotonicity implies reversibility, using an integral representation of operator convex functions and conditions on primitive divergences. Under suitable positivity and support assumptions, equality yields a recovery map, with extensions and limitations identified.

  • Recovery: Preservation of a single non-polynomial f-divergence can imply preservation of the primitive family, provided the representing measure has sufficiently large support.The integral decomposition of operator convex functions is the main mechanism behind this implication.
  • Equality conditions: Theorem 5.1 places equality, recovery by a substochastic map, modular-operator identities, and several analytic conditions into an equivalence framework.The theorem assumes supp A ≤ supp B, a substochastic map, and preservation of Tr B.
  • Equality conditions: Equality for a suitable operator convex f is connected to preservation of primitive f-divergences indexed by enough parameters.The parameter set must have cardinality at least the number of distinct eigenvalues in the relevant modular operators.
  • Recovery: A recovery map can be constructed canonically from Φ and B, and if Φ is 2-positive it can be chosen stochastic, completely positive, or n-positive under corresponding assumptions.The construction uses the adjoint map combined with support projections and preserves both A and B.
  • Limitations: The reverse implication from the canonical modular identity to equality requires additional positivity, while monotonicity itself extends to Schwarz-decomposable maps.The paper leaves open whether Schwarz decomposability alone suffices for the full reversibility implication.
  • Limitations: The support condition cannot be removed completely: a function with one-point representing-measure support may preserve an f-divergence without yielding reversibility.This provides an explicit boundary on equality-to-recovery results.

6 Distinguishability measures related to binary state discrimination

The paper relates f-divergence monotonicity to Chernoff, Hoeffding, and Rényi measures arising in binary state discrimination. It proves contraction under substochastic maps and shows that equality can imply reversibility even though Chernoff and Hoeffding distances are not themselves f-divergences.

  • Relations among measures: Rényi α-relative entropies with α ∈ [0, 1) and Hoeffding distances mutually determine each other through a convex-duality relation.The relation follows from the Legendre-Fenchel transform of the associated log-trace function.
  • Binary discrimination: Chernoff and Hoeffding distances quantify exponential error-decay rates in symmetric and asymmetric binary quantum state discrimination.The Chernoff bound concerns Bayesian error probabilities, while the Hoeffding bound constrains one error exponent and optimizes the other.
  • Monotonicity: Under a substochastic map preserving Tr Φ(B) = Tr B, both Chernoff distance and every Hoeffding distance are monotone non-increasing.If a substochastic recovery map restores A and B, these inequalities become equalities.
  • Representability: Chernoff and Hoeffding distances cannot be represented directly as f-divergences on any non-trivial finite-dimensional C∗-algebra.A commuting two-state construction shows that such a representation would force an impossible functional identity.
  • Reversibility: Equality for Chernoff distance or suitable Rényi quantities can nevertheless yield reversibility by selecting an intermediate Rényi parameter or using monotone-transform representability.For α ∈ (0,1), preservation of a Rényi α-relative entropy is sufficient under the stated 2-positivity assumptions, unlike preservation of the 0-relative entropy.

7 Error correction

The paper characterizes when preservation of suitable distinguishability measures implies reversibility of a quantum operation on a code, thereby extending quantum error-correction results to broad classes of f-divergences.

  • Setting: Quantum error correction seeks a recovery operation that reverses a noise map on a designated code of states.The paper studies when distinguishability preservation provides such a recovery criterion.
  • Reversibility criteria: For a trace-preserving 2-positive map, recoverability on a state set is equivalent to a family of distinguishability-preservation conditions, including preservation of all operator-convex f-divergences.The equivalence is formulated in Theorem 7.1.
  • Reversibility criteria: Preservation of one suitable non-polynomial operator-convex f-divergence can suffice, while primitive ϕ_t-divergences require sufficiently many parameters.The required number of parameters depends on the dimensions and the relevant representing measure.
  • Recovery construction: The recovery map can be constructed canonically from the noise map and a state with support covering the code.Under the stated positivity assumptions, the recovery map can retain corresponding positivity properties.
  • Special measures: Preservation of a single Rényi relative entropy parameter is sufficient for reversibility, unlike preservation of the continuum of Hoeffding distances.This contrasts the number of measures needed for two families of distinguishability measures.
  • Scope boundary: In non-commutative quantum systems, trace-norm distance cannot be represented as an f-divergence.The obstruction follows from the incompatibility of trace-norm distance with the f-divergence representation in this setting.

8 An integral representation for operator convex functions

The paper develops an integral representation of operator-convex functions on the positive half-line, with a non-negative measure and uniquely determined parameters, to support its f-divergence analysis.

  • Representation theorem: Every continuous operator-convex function on [0,+∞) admits an integral representation involving a real parameter, a non-negative quadratic coefficient, and a non-negative measure.The representation is characterized as an if-and-only-if condition.
  • Representation theorem: The representation parameters are uniquely determined by the function.Uniqueness includes the measure appearing in the representation.
  • Proof strategy: The proof derives the representation by applying Kraus’ theorem to a transformed function that is operator monotone.The resulting integral representation is then converted into the stated form.
  • Scope and examples: The representing kernel need not be uniquely specified pointwise when the measure has finite support, because only a weighted sum is determined.Thus the function determines the representation through the relevant aggregate rather than every individual kernel value.
  • Scope and examples: The finite-growth condition limx→+∞ f(x)/x < +∞ excludes important examples such as x log x and x^α for α ∈ (1,2].This restriction applies to the representation in the proposition requiring finite asymptotic slope.

9 Closing remarks

The closing remarks position quantum f-divergences as efficient tools for monotonicity and convexity results while emphasizing quantum-specific technical assumptions and known scope limits.

  • Scope: Quantum f-divergences encompass important distinguishability measures including relative entropy, Rényi relative entropies, and Chernoff and Hoeffding distances.They are presented as a useful, though not universal, quantum framework.
  • Technical conditions: The general monotonicity proof requires operator convexity and a decomposability condition on the map, not merely ordinary convexity and positivity.The necessity of these assumptions in full generality remains open.
  • Technical conditions: For several operational state-discrimination measures, monotonicity holds under positive trace-preserving maps whose tensor powers remain positive.The stated class includes relative entropy, Rényi relative entropies with α ∈ (0,1), and Chernoff and Hoeffding distances.
  • Technical conditions: Completely positive trace-preserving maps belong both to the tensor-power-positive class and to the decomposable class considered in the paper.The paper does not establish a broader explicit relation between these two classes.
  • Equality cases: Theorem 5.1 is presented as a broad finite-dimensional characterization of equality in the monotonicity inequality.It extends earlier equality results for several map and function classes.

A Commuting operators and the operator H¨older inequality

This section develops special monotonicity results when operators commute or when convexity alone suffices, characterizes equality cases, and derives an inverse Hölder inequality for operators.

  • Commuting operators: The commuting-operator case reduces the quasi-entropy inequality to a classical convexity argument, with strict convexity characterizing equality through constant ratios.The proof diagonalizes the operators and applies a generalized log-sum inequality.
  • Pinching and equality: For convex f, the pinching operation defined by B yields a two-step inequality relating Sf(A∥B), Sf(EB(A)∥B), and (Tr B)f(Tr A/Tr B).The first equality case requires A to commute with B, while the second requires EB(A) to be proportional to B.
  • Pinching and equality: Under supp A ≤ supp B and strict convexity, equality in the pinching inequalities occurs exactly when A commutes with B or EB(A) is a constant multiple of B, respectively.For the scalar terminal inequality, equality is likewise characterized by A being a constant multiple of B.
  • Operator Hölder inequality: The resulting trace inequalities imply that Tr A^αB^(1−α) equals (Tr A)^α(Tr B)^(1−α) only when A is a constant multiple of B.This applies for α ∈ (0,+∞) \ {1} when supp A ≤ supp B.
  • Operator Hölder inequality: Applying the trace inequality to |A|^p and |B*|^q gives the inverse Hölder inequality ∥AB∥1 ≥ ∥A∥p∥B∥q, with equality exactly when |A|^p and |B*|^q are proportional.Here p ∈ (0,1), q < 0, and 1/p + 1/q = 1, under the stated support condition.
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