PaperlayerView exact revision on arXiv

Source-linked AI summary

Different quantum f-divergences and the reversibility of quantum operations

Fumio Hiai, Milan Mosonyi

arXiv:1604.03089v4math-phcs.ITmath.ITmath.MPquant-ph

TL;DR

The paper addresses how quantum divergence families behave under quantum operations and when preservation indicates reversibility. It surveys and extends their definitions and characterizations, finding distinctions among standard, maximal, and measured f-divergences and studying α-z-Rényi divergences under invariant-argument bistochastic maps.

  • Problem

    The known monotonicity domains for α-z-Rényi divergences are incomplete, and the implications of preserving different quantum divergences for reversibility require comparison.

    Source: §1 · p. 4

  • Method

    The paper systematically reviews quantum f-divergences and their monotonicity and preservation properties, extending definitions and comparing standard, maximal, measured, and α-z-Rényi divergences.

    Source: §1 · pp. 5–6

  • Results

    For non-commuting operators, maximal f-divergences can exceed standard ones and measured f-divergences can be smaller, while the paper identifies α-z monotonicity and reversibility domains for invariant-argument bistochastic maps.

    Source: §1 · p. 6

  • Takeaways & Limitations

    Preservation of maximal f-divergences has strictly weaker reversibility consequences than preservation of standard f-divergences, with equivalence in the stated qubit-channel and commuting-output cases.

    Source: §1 · p. 5

    The alternative quantity eSf is not generally monotone under CPTP maps or jointly convex, so the paper does not treat it as a proper quantum divergence.

    Source: §3.1 · p. 13

Abstract

from arXiv · show

The concept of classical $f$-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the R\'enyi divergences. Various quantum versions of this concept, and more narrowly, the concept of R\'enyi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard $f$-divergences), Matsumoto's maximal $f$-divergences, measured $f$-divergences, and sandwiched and $\alpha$-$z$-R\'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum $f$-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum $f$-divergence by a quantum operation. In particular, we compare the standard and the maximal $f$-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum $f$-divergences are strictly different for non-commuting operators unless $f$ is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard $f$-divergences. We also study the monotonicity of the $\alpha$-$z$-R\'enyi divergences under the special class of bistochastic maps that leave one of the arguments of the R\'enyi divergence invariant, and determine domains of the parameters $\alpha,z$ where monotonicity holds, and where the preservation of the $\alpha$-$z$-R\'enyi divergence implies the reversibility of the quantum operation.

1 Introduction

The introduction frames quantum divergences around two questions: when they obey data processing, and when equality under a quantum operation implies reversibility. It surveys divergence families and compares their monotonicity and preservation properties, including for α-z-Rényi divergences.

1 Introduction

  • Known data-processing ranges differ: standard Rényi divergences are monotone for α∈[0, 2], sandwiched Rényi divergences for α∈[1/2, +∞], and the complete α-z range remains unknown.The paper studies monotonicity for α-z-Rényi divergences under a special class of bistochastic maps that leave one argument invariant.Source: §1 · pp. 4, 6
  • The introduction asks when equality in data processing means that a quantum operation is reversible on the two states, and surveys prior characterizations for several divergence families.For standard f-divergences, the paper summarizes preservation characterizations and notes an extension in which ordinary positivity of the reversion map suffices.Source: §1 · pp. 4–5
  • The paper systematically reviews quantum f-divergences, their monotonicity under quantum operations, and the implications of preserving them.It extends definitions to operator convex functions without a finite right limit at zero and establishes continuity properties for the definitions.Source: §1 · p. 5
  • Preservation of maximal f-divergences has strictly weaker reversibility consequences than preservation of standard f-divergences, though the two preservation conditions coincide for unital qubit channels and commuting outputs.The paper also constructs a qutrit channel preserving all maximal f-divergences without preserving any standard f-divergences under a stated technical condition.Source: §1 · p. 5
  • For non-commuting operators, maximal f-divergences exceed standard ones under representing-measure support conditions, and this condition can be dropped for qubits.Measured f-divergences are strictly smaller than standard f-divergences for non-commuting operators under related technical conditions.Source: §1 · p. 6

2 Preliminaries

The preliminaries establish finite-dimensional operator notation and the function-theoretic tools used to define and analyze the paper’s divergences. They also set out operator perspectives, monotone metrics, and properties of positive maps that support the later preservation results.

2.1 Notations

  • The paper works with finite-dimensional Hilbert spaces, uses B(H) for their operator algebra, and denotes positive operators, supports, traces, and left and right multiplication operators.The left and right multiplication operators commute when applied to the respective operators ρ and σ.Source: §2.1 · pp. 6–7

2.2 Operator convex and operator monotone functions

  • The paper assumes a continuous function f with specified endpoint limits and uses operator convexity and operator monotonicity, including their integral representations.Operator convexity is defined through an operator inequality, while operator monotonicity means that A ≤ B implies h(A) ≤ h(B).Source: §2.2 · p. 7

2.3 Non-commutative perspectives and operator connections

  • An operator perspective is jointly operator convex exactly when its scalar function is operator convex, and it is monotone in both arguments exactly when that function is non-negative and operator monotone.The paper also treats operator connections derived from non-negative operator monotone functions and notes that extending operator convex perspectives to non-invertible operators is nontrivial.Source: §2.3 · p. 9

2.4 Monotone metrics

  • Petz’s monotone metrics are characterized by contraction of the metric under trace-preserving maps whose adjoints are Schwarz contractions.The preliminaries associate these metrics with operator monotone functions satisfying xκ(x) = κ(x^-1).Source: §2.4 · p. 10

2.5 Positive maps

  • The preliminaries define positive, completely positive, trace-preserving, and bistochastic maps, and show that a Schwarz contraction has an algebra-valued fixed-point set when its adjoint has a positive invertible fixed point.In general, the fixed-point set need not be an algebra, and neither it nor the multiplicative domain need contain the other.Source: §2.5 · pp. 10–11

3 The standard and the maximal f-divergences

This section develops standard and maximal quantum f-divergences and studies their monotonicity and equality properties under positive maps. It characterizes when preservation or equality of these divergences holds, including reversibility results under the stated hypotheses.

3.1 Introduction to f-divergences

  • For strictly convex f, a classical f-divergence is jointly convex and stochastic-map monotone, and preservation holds exactly when the map is reversible on the input pair [34, Proposition A.3].Source: §3.1 · p. 12
  • GNS Radon–Nikodym constructions motivate standard and maximal quantum f-divergences, alongside operational maximal and measured definitions; the alternative eSf is neither CPTP-monotone nor jointly convex.Because eSf lacks both properties, the paper does not consider it further as a general quantum divergence.Source: §3.1 · pp. 12–13

3.2 Standard f-divergences

  • The standard f-divergence is extended to positive operators with arbitrary supports using a limiting definition, and Proposition 3.2 gives its spectral expression.The extension requires f to be defined on (0, +∞) with the boundary limits specified in the section.Source: §3.2 · pp. 13–14
  • For operator-convex f, the standard f-divergence is jointly convex and monotone under trace-preserving linear maps whose adjoints are Schwarz contractions.Source: §3.2 · pp. 16–17
  • Whether the monotonicity condition can be weakened from an adjoint Schwarz contraction to mere positivity is unknown, although monotonicity is noted for x^2 and Umegaki relative entropy under trace-preserving positive maps.Source: §3.2 · p. 17
  • Petz's equality conditions for 2-positive trace-preserving maps connect preservation of Umegaki relative entropy with reversibility when the map preserves σ.Source: §3.2 · pp. 17–18
  • Under the theorem’s support and 2-positivity hypotheses, reversibility is equivalent to preserving a suitable standard f-divergence and to preserving all operator-convex f-divergences.The theorem also characterizes reversibility through recovery maps, including CPTP maps that recover both states.Source: §3.2 · pp. 18–20
  • The equivalence between divergence preservation and reversibility requires conditions ensuring the divergence is finite; without a relation between the supports, the implication is unresolved in one stated case.The paper notes that the implication holds if the supports are related, or if the first state has full support; it leaves open the case where both endpoint limits of f are finite but no support relation is assumed.Source: §3.2 · p. 21

3.3 Maximal f-divergences

  • When the support condition ρ^0 ≤ σ^0 holds, the paper identifies its perspective-based divergence with Matsumoto’s maximal f-divergence.The section assumes f is operator convex on (0, +∞) to define the maximal divergence for noninvertible inputs.Source: §3.3 · p. 21
  • For operator-convex f, the regularized limit defining the maximal divergence exists for positive inputs and agrees with its original formula when both inputs are invertible.The section also gives conditions under which the operator perspective extends to noninvertible operators.Source: §3.3 · pp. 21–23
  • The maximal f-divergence is monotone under positive linear maps when the stated conditions for defining it are satisfied.The section derives this monotonicity from the corresponding operator-perspective inequality.Source: §3.3 · p. 24
  • If the map is reversible on the pair of arguments, the maximal f-divergence is preserved for every operator convex function.This implication follows from the preceding corollary.Source: §3.3 · p. 25
  • Equality of the maximal f-divergence for one non-linear operator convex function is equivalent to equality for all operator convex functions.The theorem places these equality conditions among several equivalent criteria for a positive trace-preserving map.Source: §3.3 · p. 25
  • The equivalent criteria include intertwining functional calculus and operator connections, and preservation of the square of the relative operator under Φσ.The stated conditions require the functional-calculus identity for all functions, and the operator-connection identity either for all connections or for one non-linear connection.Source: §3.3 · p. 25
  • For 2-positive maps, the theorem’s additional condition (i) is equivalent to the square-preservation condition (h) for Φσ.The passage identifies this equivalence but does not state the content of condition (i).Source: §3.3 · pp. 25, 27
  • Condition (c) is identified as a particularly easy-to-check criterion for the other conditions in Theorem 3.34.The supplied passage does not state the content of condition (c).Source: §3.3 · p. 27

4 Comparison of different f-divergences

This section compares standard, maximal, measured, and sandwiched Rényi divergences, focusing on their relations, inequalities, equality conditions, and behavior under preservation, monotonicity, and regularization. It also considers reversibility, attainment, invariance, additivity, and a Pinsker-type bound for measured f-divergences.

4.1 The relation of Sf and bSf

  • For operator-convex f, Sf(ρ ∥ σ) ≤ bSf(ρ ∥ σ); under Theorem 4.3’s technical conditions, equality implies that ρ and σ commute.Commuting inputs give equality for every f, while quadratic polynomials also give equality for all positive inputs.Source: §4.1 · pp. 28–29
  • For non-polynomial operator-convex f, the two divergences are strictly different on noncommuting qubit inputs; for f(x) = x log x, this yields S(ρ ∥ σ) < SBS(ρ ∥ σ).The qubit result applies under the stated support condition, while the relative-entropy strict inequality is given for 0 ≤ ρ ≤ σ and ρ not commuting with σ.Source: §4.1 · pp. 30–31

4.2 The relation of the preservation conditions

  • Preservation of Sf is generally stronger than preservation of bSf, which does not in general imply reversibility, even for functions satisfying the support condition.A unital qutrit pinching channel preserves bSf for every operator-convex f on [0, +∞) but strictly decreases Sf when |supp μf| ≥ 7, and is not reversible.Source: §4.2 · pp. 32–34
  • For unital qubit channels, the preservation conditions in Theorems 3.18 and 3.34 are equivalent; they are also linked when the channel’s output states commute.In the commuting-output case, the conditions of Theorem 3.34 imply those of Theorem 3.18.Source: §4.2 · pp. 34–35
  • If a 2-positive trace-preserving map has commuting outputs for every input pair, its fixed-point algebra FΦ*◦Φ equals MΦσ.This includes quantum-classical channels whose range is commutative.Source: §4.2 · p. 36

4.3 Measured f-divergence

  • Under mild technical conditions on f, equality of the measured and standard f-divergences implies that the two arguments commute.When they commute, all the inequalities relating these divergences are equalities.Source: §4.3 · p. 37
  • For operator-convex f, an optimal measurement can be chosen with at most (dim H)^2 outcomes, and the relevant measurement suprema are attained.For convex f, the projective and rank-1 projective suprema are attained; operator convexity also ensures attainment for the general measurement suprema.Source: §4.3 · pp. 37–38
  • The section relates invariance under partial isometries and isometries to monotonicity under positive trace-preserving and CPTP maps.Corollary 4.20 presents these properties as equivalent for the divergence under discussion.Source: §4.3 · p. 40
  • For fα and η(x)=x log x, the section gives a special equality result for measured and projective-measured divergences and their Rényi counterparts.The result is reviewed from [10].Source: §4.3 · p. 41
  • For α≥1/2, D*α(ρ∥σ) ≤ Dα(ρ∥σ); for positive trace-preserving maps with commuting outputs, Dα of the outputs is bounded by D*α of the inputs.The passages also state that Dα is non-increasing under such maps for α>2.Source: §4.3 · p. 42
  • The regularized measured Rényi divergence matches Dα for α≥1/2; below 1/2 this relation fails, with a strict comparison to D*α for noncommuting states.For every α>0, the passages also assert the existence of a sequence of measurements on tensor powers, though the displayed conclusion is incomplete in the supplied text.Source: §4.3 · p. 42
  • When ρ and σ commute or Dα(ρ∥σ)=+∞, the stated inequalities become equalities; otherwise they are strict, subject to specified exceptions at α=1/2 and α=1.At least one of the last two inequalities in (4.35) is strict in the non-equality case.Source: §4.3 · p. 43
  • At α=1/2, the sandwiched Rényi divergence equals −2 log F, where F is fidelity, and also equals the measured Rényi divergence.Source: §4.3 · p. 43
  • Standard and sandwiched Rényi divergences are additive, whereas measured Rényi divergences are not additive for α>1/2 when σ≠σρ; the corresponding question below 1/2 remains open.For each α>1/2 under that condition, some tensor-power size n witnesses nonadditivity.Source: §4.3 · p. 43
  • For operator-convex f on (0, +∞) with f(1)=0, Proposition 4.28 applies to every pair of density operators and generalizes the f-divergence result, strengthening the relative-entropy case.For quantum f-divergences in the sense of Section 3.1, f''(1)>0 if and only if f is nonlinear.Source: §4.3 · pp. 43–44

5 Reversibility via R´enyi divergences

This section examines when equality or preservation of Rényi divergences can imply reversibility, presenting case-specific criteria alongside counterexamples that delineate limitations for α-z Rényi divergences and max-relative entropy.

5 Reversibility via R´enyi divergences

  • Theorem 5.2 establishes monotonicity in its specified cases and characterizes equality under additional assumptions, including 2-positivity.Its equality criteria are case-specific; the theorem also gives some equivalences without assuming 2-positivity.Source: §5 · p. 46
  • For case (i), equality gives E1 ⇔ E2 ⇔ E3 ⇔ E5 when 0 ≤ σ0, but E1 ⇔ E4 when z ≠ α and σ0 ≤ 0.For case (ii), σ0 ≤ 0 gives E1 ⇔ E2 ⇔ E4 ⇔ E5; with z ≠ 1 − α and 0 ≤ σ0, it gives E1 ⇔ E3.Source: §5 · p. 46
  • For (iii) with 0 ≤ σ0, E1 ⇔ E2 ⇔ E3 ⇔ E5; for (iv) with z ≠ α − 1 and σ0 = I, E1 ⇔ E3.Without assuming Φ is 2-positive, E1 ⇔ E2 ⇔ E3 holds in (1) and (5), while E1 ⇔ E2 ⇔ E4 holds in (3).Source: §5 · p. 46
  • For cases (2), (4), and (6) of Theorem 5.2, condition (E5) does not generally follow; the remark introduces an example using Theorem 3.19’s notation with H1 = H2 = Lr.Source: §5 · p. 47
  • Corollary 5.5 gives further monotonicity and equality criteria for divergence (5.3) in three cases.The criteria include equivalences among E0, E2, E3, E4, and E5, with stated versions both with and without 2-positivity.Source: §5 · p. 47
  • For α = 2, preservation of D*2 is equivalent to reversibility on the pair for a 2-positive trace-preserving map when 0 ≤ σ0.The map need not be bistochastic.Source: §5 · p. 48

Remark 5.15

  • Preservation of fidelity does not generally imply reversibility when ρ lies in the fixed-point algebra, but it does when σ is also a fixed point, under the stated conditions.The source notes the general limitation in [52, Corollary A.9] and gives the fixed-point case as a consequence of Theorem 5.2(1).Source: §5 · p. 53

Remark 5.16

  • Preservation of the max-relative entropy does not imply reversibility, even when the second state is a fixed point of the map.The example uses invertible 3 × 3 density matrices and diagonal pinching; the stated condition establishes that the map is not reversible on the state pair.Source: §5 · p. 53

6 Closing remarks

The closing remarks extend the discussion from density operators to general positive operators and finite-dimensional C*-algebras.

6 Closing remarks

  • For operator convex f, properties of the standard and maximal f-divergences for general positive operators follow from the density-operator case by rescaling.The rescaling uses f1(x) := βf(αβ^-1x).Source: §6 · p. 54
  • Results for trace-preserving positive maps between finite-dimensional C*-algebras follow by extending the map with a trace-preserving conditional expectation.The extension is Φ composed with the conditional expectation onto the input algebra.Source: §6 · p. 54

A Extension of Lemma 2.2

The appendix records operator-convexity and operator-monotonicity criteria for perspectives and their traces, including a caveat against replacing operator monotonicity with numerical increase.

A Extension of Lemma 2.2

  • The appendix invokes Kraus’ theorem for the equivalence between operator convexity of ef and that of f.The passage states that either follows from the other.Source: A Extension of Lemma 2.2 · pp. 54–55
  • Joint operator convexity of Pf is equivalent to convexity of its trace perspective in either variable and operator convexity of ef.The proof cites established implications and uses Lemma 2.1 to connect the criteria.Source: A Extension of Lemma 2.2 · pp. 54–55
  • Proposition A.2 equates joint operator monotonicity of Ph with trace monotonicity and criteria involving operator monotonicity of h and eh.The listed conditions also require numerical increase of the complementary function.Source: A Extension of Lemma 2.2 · pp. 55–56
  • For operator-monotone h, h is non-negative exactly when eh is numerically increasing on (0, +∞).This is Lemma A.3, used to establish equivalences in Proposition A.2.Source: A Extension of Lemma 2.2 · pp. 55–56
  • Numerical increase alone cannot replace operator monotonicity in Proposition A.2 or Lemma A.3.For f(t) := t2 + 1, f is non-negative and numerically increasing, yet ef is neither increasing nor decreasing.Source: A Extension of Lemma 2.2 · p. 56

B Examples for FΦ and MΦ

The examples show how unitary and stochastic maps shape the fixed-point sets FΦ and MΦ, including cases where those sets differ sharply.

B Examples for FΦ and MΦ

  • For a unitary conjugation, the inclusion between the fixed-point sets is strict unless U is a scalar multiple of the identity.Source: B Examples for FΦ and MΦ · p. 56
  • The stochastic-matrix construction gives a unital completely positive map, and it is trace-preserving exactly when the matrix is bistochastic.Source: B Examples for FΦ and MΦ · p. 57
  • If a stochastic matrix has a strictly positive row, then MΦ consists only of scalar multiples of the identity.If none of its columns is zero, every element of MΦ is diagonal in the specified basis.Source: B Examples for FΦ and MΦ · p. 57
  • In Example B.3, MΦ consists of scalar multiples of the identity, while I and A are fixed points and A2 is not, so FΦ is not an algebra.The example also states that Φ cannot have a faithful invariant state.Source: B Examples for FΦ and MΦ · p. 58

C Example for eSf(̺ ∥σ)

The section constructs a nonlinear operator-convex function whose standard quantum f-divergence is neither monotone increasing nor monotone decreasing under CPTP maps. It also derives a related failure of convexity.

C Example for eSf(̺ ∥σ)

  • When the input operators commute and the positive trace-preserving map preserves their commutativity, classical f-divergence monotonicity applies, with strict inequality possible.The strictness is attributed to [34, Proposition A.3].Source: C Example for eSf(̺ ∥σ) · p. 58
  • For every δ > 0, fδ(x) = 1 − x + δ(1 − x)^2 is nonlinear and operator convex, yet its standard quantum f-divergence is neither increasing nor decreasing under CPTP maps.The nonmonotonicity conclusion is established using the conditions described for t ∈ (0, 1) \ {1/2} and sufficiently small ε and δ.Source: C Example for eSf(̺ ∥σ) · pp. 58–59
  • For any δ ∈ (0, δt), the function ̺ ↦ eSfδ(̺ ∥σt) is not convex on invertible density operators.This is the additional consequence stated in Remark C.1.Source: C Example for eSf(̺ ∥σ) · p. 59

D Continuity properties of the standard f-divergences

This section establishes continuity properties of standard f-divergences under positive perturbations when f is operator convex. It also gives examples showing that convexity alone does not ensure the corresponding continuity.

D Continuity properties of the standard f-divergences

  • If f(0+) and f ′(+∞) are finite, continuity of the perspective function yields the stated limit in (3.10).This continuity holds on [0, +∞) × [0, +∞).Source: D Continuity properties of the… · p. 59
  • For operator-convex f, adding the same positive sequence Ln to both arguments preserves the limiting standard f-divergence, as shown by matching upper and lower bounds.The upper bound follows from joint convexity, and the lower bound is obtained using the integral expression and Fatou’s lemma.Source: D Continuity properties of the… · pp. 59–60
  • Convexity without operator convexity is insufficient for continuity when the same positive perturbations are added to both arguments.Example D.1 gives a convex, non-operator-convex f for which the perturbed divergences do not converge to the original value.Source: D Continuity properties of the… · p. 61
  • For operator-convex f with finite f(0+), perturbing only σ by positive Ln yields convergence; the upper bound adds f(0+) Tr Ln.The lower-bound argument reduces to the f2 term after the other terms converge.Source: D Continuity properties of the… · p. 62
  • Convexity alone also fails to guarantee the one-sided continuity result; Example D.3 uses f(x) = x^α with α > 3.The example has f(0+) = 0 and 0 ≤ σ0.Source: D Continuity properties of the… · p. 62

E Proof of Proposition 3.26

The proof establishes Proposition 3.26 by deriving convergence results for operator-convex functions and then decomposing a general such function into two parts. It combines the corresponding limits to obtain the final conclusion.

E Proof of Proposition 3.26

  • For part (i), the proof uses the operator-monotone function derived from f to obtain the needed convergence results.It forms g(x) = (f(x) − f(1))/(x − 1), then uses h(x) = g(x) − g(0+) as a nonnegative operator-monotone function.Source: E Proof of Proposition 3.26 · pp. 63, 65
  • Part (ii) follows from part (i) by Lemma 2.1..Source: E Proof of Proposition 3.26 · p. 65
  • For part (iii), the proof decomposes f as f0 + f1 and applies parts (i) and (ii), then combines the resulting limits.The conclusion follows by adding the two obtained expressions.Source: E Proof of Proposition 3.26 · p. 65
Loading 1604.03089v4…