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arXiv:1412.3676

On maximization of measured $f$-divergence between a given pair of quantum states

Keiji Matsumoto

Published Dec 11, 2014• 2 citations
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Abstract

This paper deals with maximization of classical $f$-divergence between the distributions of a measurement outputs of a given pair of quantum states. $f$-divergence $D_{f}$ between the probability density functions $p_{1}$ and $p_{2}$ over a discrete set is defined as $D_{f}( p_{1}||p_{2}) :=\sum_{x}p_{2}(x) f\left(p_{1}(x)/p_{2}( x) \right) $. For example, Kullback-Leibler divergence and Renyi type relative entropy are well-known examples with good operational meanings. Thus, finding the maximal value $D_{f}^{\min}$ of measured measured $f$-divergence is also an interesting question. But so far the question is solved only for very restricted example of $f$. \ The purposes of the present paper is to advance the study further, by investigating its properties, rewriting the maximization problem to more tractable form, and giving closed formulas of the quantity in some special cases.

DOI
10.48550/arxiv.1412.3676

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On maximization of measured fff-divergence between a given pair of quantum states
This paper deals with maximization of classical fff-divergence between the distributions of a measurement outputs of a given pair of quantum states. fff-divergence DfD_{f}Df​ between the probability density functions p1p_{1}p1​ and p2p_{2}p2​ over a discrete set is defined as Df(p1∣∣p2):=∑xp2(x)f(p1(x)/p2(x))D_{f}( p_{1}||p_{2}) :=\sum_{x}p_{2}(x) f\left(p_{1}(x)/p_{2}( x) \right) Df​(p1​∣∣p2​):=∑x​p2​(x)f(p1​(x)/p2​(x)). For example, Kullback-Leibler divergence and Renyi type relative entropy are well-known examples with good operational meanings. Thus, finding the maximal value Dfmin⁡D_{f}^{\min}Dfmin​ of measured measured fff-divergence is also an interesting question. But so far the question is solved only for very restricted example of fff. \ The purposes of the present paper is to advance the study further, by investigating its properties, rewriting the maximization problem to more tractable form, and giving closed formulas of the quantity in some special cases.
This work presents an upper-bound for the maximum value that the Kullback-Leibler (KL) divergence can reach for a newly introduced class of probability distributions, called quantum distributions (QD). In particular, the aim is to find a discrete distribution UUU which maximize the finite KL divergence from a given PPP under the assumption that PPP and UUU have been generated by distributing a fixed discretized quantity. The theoretical findings are used for proposing a notion of normalized KL divergence that is empirically shown to behave differently from already known measures.
This work presents an upper-bound to value that the Kullback-Leibler (KL) divergence can reach for a class of probability distributions called quantum distributions (QD). The aim is to find a distribution UUU which maximizes the KL divergence from a given distribution PPP under the assumption that PPP and UUU have been generated by distributing a given discrete quantity, a quantum. Quantum distributions naturally represent a wide range of probability distributions that are used in practical applications. Moreover, such a class of distributions can be obtained as an approximation of any probability distribution. The retrieving of an upper-bound for the entropic divergence is here shown to be possible under the condition that the compared distributions are quantum distributions over the same quantum value, thus they become comparable. Thus, entropic divergence acquires a more powerful meaning when it is applied to comparable distributions. This aspect should be taken into account in future developments of divergences. The theoretical findings are used for proposing a notion of normalized KL divergence that is empirically shown to behave differently from already known measures.
Maximal quantum fff-divergences, defined via the commutant Radon--Nikodym derivative, form a fundamental class of distinguishability measures for quantum states associated with operator convex functions. In this paper, we study the optimization of maximal quantum fff-divergences along unitary orbits of two quantum states. For any operator convex function f:(0,+∞)→Rf:(0,+\infty)\to\mathbb{R}f:(0,+∞)→R, we determine the exact minimum and maximum of U⟼S^f(ρ∥U∗σU) U \longmapsto \widehat S_f(ρ\|U^*σU) U⟼Sf​(ρ∥U∗σU) over the unitary group, and derive explicit spectral formulas for these extremal values together with complete characterizations of the unitaries that attain them. Our approach combines the integral representation of operator convex functions with majorization theory and a unitary-orbit variational method. A key step is to show that any extremizer must commute with the reference state, which reduces the noncommutative optimization problem to a spectral permutation problem. As a consequence, the minimum is achieved by pairing the decreasing eigenvalues of ρρρ and σσσ, while the maximum corresponds to pairing the decreasing eigenvalues of ρρρ with the increasing eigenvalues of σσσ. Hence, the range of the maximal quantum fff-divergence along the unitary orbit is exactly the closed interval determined by these two extremal configurations. Finally, we compare our results with recent unitary-orbit optimization results for quantum fff-divergences defined via the quantum hockey-stick divergence, highlighting fundamental structural differences between the two frameworks. Our findings extend earlier extremal results for Umegaki, Rényi, and related quantum divergences, and clarify the distinct operator-theoretic nature of maximal quantum fff-divergences.
Abstract The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f -divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz–Rényi relative entropies. In this paper, I introduce the optimized quantum f -divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz’s original approach. Interestingly, the sandwiched Rényi relative entropies are particular examples of the optimized f -divergence. Thus, one benefit of this paper is that there is now a single, unified approach for establishing the data processing inequality for both the Petz–Rényi and sandwiched Rényi relative entropies, for the full range of parameters for which it is known to hold. This paper discusses other aspects of the optimized f -divergence, such as the classical case, the classical-quantum case, and how to construct optimized f -information measures.
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 fff-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 fff-divergences), Matsumoto's maximal fff-divergences, measured fff-divergences, and sandwiched and α\alphaα-zzz-R\'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum fff-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum fff-divergence by a quantum operation. In particular, we compare the standard and the maximal fff-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum fff-divergences are strictly different for non-commuting operators unless fff is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard fff-divergences. We also study the monotonicity of the α\alphaα-zzz-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\alpha,zα,z where monotonicity holds, and where the preservation of the α\alphaα-zzz-R\'enyi divergence implies the reversibility of the quantum operation.