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-Szkoła 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.
Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.
We introduce the geodesic quantum
f-divergences
Dft,
0≤t≤1, obtained from the affine-invariant geodesic between the standard and maximal relative modular operators. They reduce to the classical
f-divergence for commuting states. The logarithmic generator yields geodesic relative entropies joining the Umegaki and Belavkin-Staszewski entropies, while the power generators yield
(t,α)-Rényi divergences joining the Petz and geometric families.
Our first main result is data processing of
Dft for every finite operator-convex generator
f and every
t∈[0,1]. In particular, this gives DPI for the geodesic relative entropies and for the
(t,α)-Rényi divergences when
0<α<1 or
1<α≤2. Our second main result identifies equality in the DPI: for invertible states, every equality-determining operator-convex generator has, at each nonmaximal parameter
0≤t<1, exactly the Petz sufficiency class; at
t=1, this changes to the generally larger maximal, or BS, class, which also coincides with the equality class of the divergences associated with the quadratic generator for every
t. Our third main result is the corresponding collapse of invertible geodesic quantum Markov chains: for each of the three ordered conditional-mutual-information constructions, the
t-quantum Markov chains are precisely the quantum Markov chains for
0≤t<1, whereas at
t=1 they are the invertible BS quantum Markov chains, a class that can be strictly larger.
We also determine parameter-monotonicity regimes and the intersections with the
(α,z) family. We prove strengthened data-processing and reconstruction estimates; we compare the three conditional orientations; we establish continuity bounds under positive lower-eigenvalue assumptions together with complementary discontinuity results; and we give a capacity-per-unit-cost interpretation.
Abstract We introduce a fully quantum notion of entropy production based on the non-commutative extension of the classical log-ratio between forward and reverse processes. Given a pair of quantum objects associated with the forward and reverse descriptions, we define a Hermitian entropy-production operator whose expectation value is non-negative and equal to the Belavkin–Staszewski relative entropy. The operator satisfies exact integral and detailed fluctuation theorems without requiring commutativity. We then specialize this construction to the case in which the forward process is described by a single quantum channel and the reverse process is defined inferentially, through Bayesian retrodiction relative to a prior state, with the Petz transpose map as the Bayesian inverse. In this setting, the relevant quantities can be evaluated explicitly, leading to a number of natural structural and physical properties. The framework recovers the classical formula in the commutative limit, yields explicit expressions for the average entropy production, and clarifies where the fully quantum case departs from standard thermodynamic expectations.
The minimum change principle provides an information-theoretic characterization of the Bayes reversal channel in classical probability theory and has recently been proposed as a framework for extending Bayes' rule to quantum information theory. Using quantum relative entropy, we investigate a minimum change principle for the setting of quantum statistical inference. Specifically, we consider a forward process based on a classical-to-quantum preparation channel and a reverse process based on a quantum-to-classical measurement channel. We establish a closed-form characterization of measurements that are optimal for this principle, and this optimal measurement can be found via a dual formulation involving a single unconstrained Hermitian variable. This perspective allows us to recover some notable measurements within the same framework, including pretty good measurements and Fermi-Dirac thermal measurements, and we use it to discover a novel family that we call softmin thermal measurements. We further show that softmin thermal measurements arise as optimal solutions to entropy-regularized semidefinite optimization problems, demonstrating that they play a role for measurements analogous to that of thermal states in statistical mechanics. Finally, we prove an additivity property for the relative-entropy minimum change principle and investigate the performance of Fermi-Dirac thermal measurements for quantum hypothesis testing.
We study quantum measurement retrodiction using the principle of minimum change. For quantum-to-classical measurement channels, we show that all standard quantum divergences select the same retrodictive update, yielding a unique and divergence-independent quantum Bayesian inverse for any POVM and prior state. Using this update, we construct a symmetric joint distribution for pairs of POVMs and introduce the mutual retrodictability, for which we also derive a general upper bound that depends only on the prior state and holds for all measurements. This structure leads to two retrodictive entropic uncertainty relations, expressed directly in terms of the prior state and the POVMs, but valid independently of the retrodictive framework and fully compatible with the conventional operational interpretation of entropic uncertainty relations. Finally, we benchmark these relations numerically and find that they provide consistently tighter bounds than existing entropic uncertainty relations over broad classes of measurements and states.
We study the Hoeffding regime of composite quantum hypothesis testing, in which each hypothesis is specified by a sequence of sets of quantum states. We establish quantum Hoeffding bounds under a set of structural assumptions, orthogonal to those of our previous framework. A notable consequence is the direct operational interpretation of the reverse sandwiched Renyi divergence for
α∈(0,1): for the task of discriminating a thermal equilibrium state from a probe state subject to unknown dephasing in the energy eigenbasis, with free Hamiltonian evolution as a special case, the optimal Hoeffding exponent is given exactly by this divergence evaluated on a single copy of the system. The same task in the Stein regime is governed by the reverse quantum relative entropy, providing its operational interpretation as well. This behavior contrasts both with the simple independent and identically distributed (i.i.d.) setting, where the Petz Renyi divergence and the Umegaki relative entropy govern the Hoeffding and Stein exponents, respectively, and with many composite settings, where only regularized many-copy formulas are available. This finding reveals that passing from simple to composite hypotheses can fundamentally change which quantum divergence determines the operational limits of discrimination, and suggests a new avenue for seeking operational interpretations of quantum divergences by lifting simple hypotheses to richer composite scenarios.