The main result in this paper shows that the quantum [Formula: see text]-divergence of two states is equal to the classical [Formula: see text]-divergence of the corresponding Nussbaum–Szkoła distributions. This provides a general framework for studying certain properties of quantum entropic quantities using the corresponding classical entities. The usefulness of the main result is illustrated by obtaining several quantum [Formula: see text]-divergence inequalities from their classical counterparts. All results presented here are valid in both finite and infinite dimensions and hence can be applied to continuous variable systems as well. A comprehensive review of the instances in the literature where Nussbaum–Szkoła distributions are used, is also provided in this paper.
The main result in this paper shows that the quantum [Formula: see text]-divergence of two states is equal to the classical [Formula: see text]-divergence of the corresponding Nussbaum–Szkoła distributions. This provides a general framework for studying certain properties of quantum entropic quantities using the corresponding classical entities. The usefulness of the main result is illustrated by obtaining several quantum [Formula: see text]-divergence inequalities from their classical counterparts. All results presented here are valid in both finite and infinite dimensions and hence can be applied to continuous variable systems as well. A comprehensive review of the instances in the literature where Nussbaum–Szkoła distributions are used, is also provided in this paper.
Quantum f-divergences via Nussbaum-Szkoła Distributions in Semifinite von Neumann Algebras
In this article, we prove that the quantum f-divergence between two normal states on a semifinite von~Neumann algebra is equal to the classical f-divergence between two corresponding classical states, which are called Nussbaum-Szkoła distributions. This result has been proved by the second named author and T.C.~John for normal states on the von~Neumann algebra B(H) of all bounded operators on a Hilbert space H. We extend their result for normal states on any semifinite von~Neumann algebra, not only B(H).
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.
The concept of classical [Formula: see text]-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the Rényi divergences. Various quantum versions of this concept, and more narrowly, the concept of Rényi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz’ quasi-entropies (standard [Formula: see text]-divergences), Matsumoto’s maximal [Formula: see text]-divergences, measured [Formula: see text]-divergences, and sandwiched and [Formula: see text]-[Formula: see text]-Rényi divergences. In this paper, we give a systematic overview of the various concepts of quantum [Formula: see text]-divergences, with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum [Formula: see text]-divergence by a quantum operation. In particular, we compare the standard and the maximal [Formula: see text]-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum [Formula: see text]-divergences are strictly different for non-commuting operators unless [Formula: see text] is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard [Formula: see text]-divergences. We also study the monotonicity of the [Formula: see text]-[Formula: see text]-Rényi divergences under the special class of bistochastic maps that leave one of the arguments of the Rényi divergence invariant, and determine domains of the parameters [Formula: see text] where monotonicity holds, and where the preservation of the [Formula: see text]-[Formula: see text]-Rényi divergence implies the reversibility of the quantum operation.
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.
Quantum hypothesis testing concerns the discrimination between quantum states. This paper introduces novel lower and upper bounds for asymmetric quantum hypothesis testing based on the Nussbaum-Szkoła mapping. Together, these bounds provide a unified recovery of both converse and achievability results across all major asymptotic regimes, including large-, moderate-, and small-deviations. Unlike existing bounds, which either rely on technically involved information-spectrum arguments or suffer from fixed prefactors and limited applicability in the non-asymptotic regime, the proposed bounds arise from a unified framework and enable, in some cases, the direct use of classical results. It is further demonstrated that these bounds provide accurate approximations to the optimal quantum error trade-off function at small blocklengths. Numerical comparisons with existing bounds, including those based on fidelity and information spectrum methods, highlight their improved tightness.
We obtain formulas for Petz–Rényi and Umegaki relative entropy from the idea of distribution of a positive self-adjoint operator. Classical results on Rényi and Kullback–Leibler divergences are applied to obtain new results and new proofs for some known results about Petz–Rényi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz–Rényi [Formula: see text]-relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.