Pinsker's inequality sets a lower bound on the Umegaki divergence of two quantum states in terms of their trace distance. In this work, we formulate corresponding estimates for a variety of quantum and classical divergences including
f-divergences like Hellinger and
χ2-divergences as well as Rényi divergences and special cases thereof like the Umegaki divergence, collision divergence, max divergence. We further provide a strategy on how to adapt these bounds to smoothed divergences.
Recently, a new definition for quantum
f-divergences was introduced based on an integral representation. These divergences have shown remarkable properties, for example when investigating contraction coefficients under noisy channels. At the same time, many properties well known for other definitions have remained elusive for the new quantum
f-divergence because of its unusual representation. In this work, we investigate alternative ways of expressing these quantum
f-divergences. We leverage these expressions to prove new properties of these
f-divergences and demonstrate some applications. In particular, we give a new proof of the achievability of the quantum Chernoff bound by establishing a strengthening of an inequality by Audenaert et al. We also establish inequalities between some previously known Renyi divergences and the new Renyi divergence. We further investigate some monotonicity and convexity properties of the new
f-divergences, and prove inequalities between these divergences for various functions.
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 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.
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
α-
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
α-
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
α,z where monotonicity holds, and where the preservation of the
α-
z-R\'enyi divergence implies the reversibility of the quantum operation.
Data-processing inequalities capture the phenomenon that two probability distributions can only become less distinguishable under any common post-processing. For more fine-grained inequalities, one turns to strong data-processing inequality (SDPI) constants, which give the strongest inequalities for a given channel and reference state for a fixed measure of distinguishability. These quantities have been used to quantify the rate at which time-homogeneous Markov chains contract towards a fixed point both in the classical and quantum setting. In this work, we establish that quantum
f-divergences satisfy a local reverse Pinsker inequality, which implies the asymptotic contraction rate of a primitive channel to its stationary state is upper bounded by the SDPI constant of any non-commutative
χ2-divergence. Using quantum-detailed balance, we establish a sufficient condition for these bounds to be tight. Finally, we apply these results to Petz, Matsumoto, and Hirche-Tomamichel
f-divergences, establishing new and strengthening previously known results.