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Quantum f-divergences from retrodictive tests: a unifying classical variational principle

Francesco Buscemi

arXiv:2610.29696v2quant-phcs.ITmath-phmath.ITmath.MP

TL;DR

Classical f-divergences can have inequivalent quantum extensions, raising the question of whether they share an operational variational principle. The paper introduces reference-dependent retrodictive tests and proves that one spectral implementation minimizes the classical divergence for all operator-convex generators. Reference choices recover standard, maximal, and interpolating quantum divergences, and the optimization also extends to channels.

  • Problem

    The same classical f-divergence has inequivalent extensions to quantum states, and the paper asks whether they can arise from a common operational variational principle.

    Source: §1 · p. 1

  • Method

    The paper minimizes classical f-divergence between forward and reverse preparation-measurement processes coupled by reference-relative Petz transposition and constrained by prescribed quantum averages.

    Source: §1 · p. 2

  • Results

    The same spectral implementation attains the minimum over admissible implementations for every operator-convex f, while reference choices recover standard and maximal quantum divergences.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Changing the reference yields different quantum extensions of the same classical divergence, and the optimization extends to quantum channels, characterizing entropy production for channels with faithful Choi operator.

    Source: §1 · p. 2

    The optimization theorem considers faithful finite-dimensional states and finite admissible implementations, with outcome counts arbitrary but finite.

    Source: §3 · p. 6

Abstract

from arXiv · show

The same classical f-divergence admits inequivalent extensions to quantum states. The question is whether these extensions can be obtained from a common classical variational principle with an operational interpretation. Here we generalize Matsumoto's reverse-test optimization by introducing retrodictive tests relative to a reference state. A retrodictive test compares a forward and a reverse stochastic process, realized by quantum preparation and measurement, related by Petz transposition relative to the chosen reference, and subject to prescribed quantum averages. Equivalently, it compares two reverse tests sharing a reference output, whose labels are coupled by retrodiction. For faithful states in finite dimension, we prove that the same spectral implementation attains the minimum classical f-divergence over all admissible implementations simultaneously for every operator-convex f. A flat reference yields the standard quantum f-divergences, thereby giving the Nussbaum-Szko{\l}a distributions a variational characterization. Choosing either prescribed state as reference recovers maximal quantum f-divergences and Matsumoto's reverse-test optimization, while power references interpolate between the standard and maximal divergences along the geodesic family. In the relative-entropy case, this gives the passage from Umegaki to Belavkin-Staszewski relative entropy by changing the reference state. The same optimization extends to arbitrary quantum channels and, for channels with faithful Choi operator, gives a variational characterization of quantum entropy production.

1 Introduction: the operational question

The paper asks how inequivalent quantum extensions of a classical divergence can arise from one operational principle. It introduces reference-dependent retrodictive tests and proves a shared spectral optimizer, with applications to quantum divergences and channels.

1 Introduction: the operational question

  • A retrodictive test compares forward and reverse stochastic processes realized by quantum preparation and measurement, with their pairing set by Petz transposition relative to reference γ.The prescribed averages are ρ and σ, while γ determines the retrodictive relation between implementations.Source: §1 · p. 2
  • The optimization minimizes classical f-divergence between forward and reverse processes over implementations constrained by prescribed averages and a reference state.Its minimum measures distinguishability that cannot be removed by changing the implementation while respecting those constraints.Source: §1 · p. 2
  • The same spectral decompositions of two relative density operators attain the minimum simultaneously for every operator-convex f.For arbitrary reference states the minimum is a Petz quasi-entropy; a flat reference gives the standard quantum f-divergence and Nussbaum–Szkoła distributions, while γ=ρ or σ recovers maximal divergences and Matsumoto’s reverse-test optimization.Source: §1 · p. 2
  • Power references interpolate between the flat-reference and endpoint cases, giving geodesic quantum f-divergences; for f(u)=u log u, the specializations include Umegaki, Belavkin–Staszewski, and geodesic relative entropies.Changing the reference while holding f fixed produces different quantum extensions of the same classical divergence.Source: §1 · p. 2
  • The variational characterization also proves separate convexity at fixed reference and a stronger logarithmic mixture bound for relative entropy, and extends to channels with Petz-transposed reverse processes.For channels with faithful Choi operator, it gives a classical variational characterization of average entropy production introduced in [BBS26].Source: §1 · p. 2

2 Retrodictive tests with prescribed quantum averages

A retrodictive test is built from forward and reverse processes that share a reference coupling and obey prescribed quantum-average constraints. Its admissible implementations can be parameterized by weighted POVM resolutions, with Petz transposition furnishing the retrodictive pairing.

2 Retrodictive tests with prescribed quantum averages

  • An admissible implementation uses finite families of prepared states and strictly positive reference distributions, with preparation distributions that may contain zero entries.The states ρ, σ, and γ are faithful density operators on a finite-dimensional Hilbert space.Source: §2 · p. 3
  • The freedom in an implementation is equivalent to choosing two finite weighted POVM resolutions of fixed relative density operators; the POVMs need not be projective, commuting, or jointly measurable.Weights may vanish even when the fixed operators are positive.Source: §2 · p. 3
  • The preparation channels associated with measurement outcomes are Petz transposes relative to γ, pairing preparation and measurement stages across the two processes.Figure 1 distinguishes classical labels from quantum states and depicts matching colors for Petz-transposed components.Source: §2 · pp. 3–4, Figure 1 · p. 5
  • Petz transposition makes the reverse transition the Bayesian inverse of the forward transition at reference law g, while the reverse starting law is separately constrained to realize σ.The record retains both labels in the order (i, j), and the forward and reverse processes are two reverse tests sharing reference output γ.Source: §2 · p. 4

3 Spectral optimality

For faithful finite-dimensional states, spectral resolutions of the relative density operators minimize the classical f-divergence over admissible implementations. The result applies simultaneously to operator-convex generators and recovers several quantum divergences under particular references.

3 Spectral optimality

  • The spectral POVMs of the relative density operators attain the minimum simultaneously for every operator-convex generator f with f(1)=0.Two applications of operator Jensen establish the lower bound; choosing the spectral resolutions makes both inequalities equalities.Source: §3 · pp. 6–7
  • The optimizer depends on ρ, σ, and γ but not on f, although the minimum value generally does.The theorem optimizes over finite admissible implementations, with arbitrary finite outcome counts.Source: §3 · pp. 6, 8
  • For arbitrary reference γ, the minimum is a Petz quasi-entropy; at γ=I/d it becomes the standard quantum f-divergence.The flat reference also supplies a variational characterization of the Nussbaum–Szkoła distributions.Source: §3 · p. 8
  • For the relative-entropy generator, the minimum equals Tr[γB log B] − Tr[σ log A], with A=γ^-1/2ργ^-1/2 and B=γ^-1/2σγ^-1/2.This is DBS(σ∥γ) − Tr[σ log A], and the variational characterization implies that the expression is nonnegative.Source: §3 · p. 8
  • The minimum is zero exactly when ρ=σ and [ρ,γ]=0; equal averages alone need not make the forward and reverse record laws equal.If ρ=σ but [ρ,γ]≠0, every admissible implementation has strictly positive relative entropy between its processes.Source: §3 · pp. 8–9

4 Quantum divergences selected by the reference

The chosen reference state selects which quantum extension of a classical f-divergence the retrodictive-test minimum realizes. Flat, endpoint, and power references yield standard, maximal, and geodesic divergences, respectively.

4 Quantum divergences selected by the reference

  • For a general reference, the minimum is reference-dependent, and the complete record laws need not come from one preparation channel producing both prescribed states.The shared-device reverse-test correspondence applies at the endpoint references.Source: §4 · p. 9, §4.2 · pp. 10, 12

4.1 Flat reference: standard divergences and Nussbaum–Szkoła distributions

  • With γ = I/d, the spectral records give the standard quantum f-divergence and Nussbaum–Szkoła distributions, minimizing every operator-convex classical f-divergence.For f(u) = u log u, the minimum is the Umegaki relative entropy.Source: §4.1 · pp. 9–10

4.2 Endpoint references: maximal divergences and Matsumoto’s reverse tests

  • At γ = ρ or γ = σ, the minimum is the maximal quantum f-divergence and the optimization recovers Matsumoto’s reverse test.At γ = ρ, its canonical record laws are p and q, with likelihood ratios q_j/p_j given by eigenvalues of B.Source: §4.2 · pp. 10–11

4.3 Power references: geodesic quantum f-divergences

  • Across the power family, changing the reference changes the retrodictive relation; at t = 0 the records are Nussbaum–Szkoła distributions, while at t = 1 they reduce to the endpoint reverse test.The forward measurement and reverse preparation ensemble can remain fixed throughout the family.Source: §4.3 · pp. 12–13
  • For a power reference γ_t, the minimum is the geodesic quantum f-divergence, connecting the flat-reference and endpoint constructions.For relative entropy, this gives the passage from Umegaki to Belavkin–Staszewski relative entropy.Source: §4.3 · pp. 13–14

5 Convexity from mixing admissible implementations

At a fixed reference, the paper derives separate convexity of the optimized divergence by combining optimal implementations. In the relative-entropy case, varying the mixture weights yields a stronger logarithmic mixture bound.

5 Convexity from mixing admissible implementations

  • Combining component optimizers with a common weighted POVM resolution on the fixed-state side yields an admissible implementation for the mixture.The component label is retained in both records; for convexity in ρ, the forward label has weights p while the reverse label can use weights q.Source: §5 · pp. 14–15
  • The variational proof gives separate convexity at fixed γ and, for relative entropy, the stronger conclusion that exp[−V(ρ, σ; γ)] is concave in ρ.The logarithmic mixture bound also applies to Umegaki, geodesic, and Belavkin–Staszewski relative entropies with their respective references held fixed.Source: §5 · p. 14
  • For relative entropy, choosing the reverse component weights q yields the logarithmic mixture bound through the classical chain rule.The variational construction realizes the classical cost of the bound by combining optimal implementations and varying those probabilities.Source: §5 · pp. 15–16

6 The same optimization with a quantum channel

The spectral-optimality result extends to a prescribed quantum channel between preparation and measurement, with its Petz transpose used in the reverse process. For channels with faithful Choi operator, the relative-entropy variational formula characterizes average entropy production.

6 The same optimization with a quantum channel

  • With a channel between preparation and measurement, the spectral implementation minimizes the classical f-divergence simultaneously for every operator-convex f.The result assumes faithful input states and a faithful output state on the support of η = E(γ).Source: §6 · pp. 16–17
  • The forward process prepares an input ensemble, applies E, and measures the output, while the reverse process uses the Petz transpose and measures the input.Both processes report the pair (i, j), and the reference coupling has marginals g and h.Source: §6 · pp. 16–17
  • For channels with faithful Choi operator, the second relative-entropy minimum agrees with average entropy production; the variational formulas hold for every channel under the theorem’s support assumptions.For the identity channel, the formulas reduce to the earlier state formulas.Source: §6 · p. 17
  • The entropy-production identification does not extend to every channel: for a unitary channel, the cited entropy production vanishes, while the identity-channel minimum can be positive when ρ ≠ σ.The paper distinguishes this comparison from the variational formulas, which hold for every channel under the stated support assumptions.Source: §6 · p. 17

7 Discussion: comparing reverse tests through a common reference

The discussion interprets the optimization as a comparison of two reverse tests sharing a reference output, with Petz retrodiction coupling their labels. The same spectral choices minimize this coupled comparison, while the reference determines which quantum extensions the task realizes.

7 Discussion: comparing reverse tests through a common reference

  • The admissible implementation uses two reverse tests sharing reference output γ, with Petz transposition pairing their preparation devices and coupling their labels through J.The tests satisfy C(a)=ρ, D(b)=σ, and C(g)=D(h)=γ.Source: §7 · p. 18
  • The separate divergences compare each record law with J, but the direct comparison Df(R∥F) depends on how J pairs their likelihood ratios.Thus the separate divergence values alone cannot generally recover the coupled comparison.Source: §7 · p. 18
  • The spectral resolutions of A and B minimize the coupled comparison simultaneously for every operator-convex generator, beyond their separate reverse-test minima.Replacing either weighted POVM resolution by its spectral resolution cannot increase the coupled divergence, regardless of the resolution chosen on the other side.Source: §7 · p. 18
  • At an endpoint reference, one canonical test is trivial and the comparison reduces to a single reverse test; the channel extension uses input and output references γ and η=E(γ).The transmitted system passes through E or its Petz transpose.Source: §7 · p. 18
  • Changing the reference changes how the classical descriptions are compared while keeping the prescribed averages and classical divergence fixed, connecting standard, maximal, and geodesic divergences.These are different reference choices within the same operational task.Source: §7 · p. 19
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