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arXiv:1412.3680

On the condition of conversion of classical probability distribution families into quantum families

Keiji Matsumoto

Published Dec 11, 2014• 4 citations
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Abstract

The purpose of the paper is to study the condition for a probability distribution family to a quantum state family. This is an (relatively) easy example of quantum version of "comparison of statistical experiments", which had turned out to supply deep insight into the foundation of classical and quantum statistics \cite{Strasser}\cite{Torgersen}. It turns out use of maximal quantum $f$-divergence is useful in characterizing the classical-quantum transformability.

DOI
10.48550/arxiv.1412.3680

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The purpose of the paper is to study the condition for a probability distribution family to a quantum state family. This is an (relatively) easy example of quantum version of "comparison of statistical experiments", which had turned out to supply deep insight into the foundation of classical and quantum statistics \cite{Strasser}\cite{Torgersen}. It turns out use of maximal quantum fff-divergence is useful in characterizing the classical-quantum transformability.
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In this PhD thesis, we take an axiomatic approach to studying multivariate divergences, which are measures of the information content of a tuple of classical or quantum states. These divergences are studied in the context of large-sample and catalytic convertibility of tuples of states. This type of convertibility is characterized by the ordering of an axiomatically defined family of divergences. In this thesis, we identify this family of divergences explicitly for two scenarios, one classical and one quantum. In the classical setting, we show that the relevant divergences are certain generalizations of the bivariate Rényi divergences. Unlike previous studies, we do not require the distributions to have the same support, yielding interesting new divergences. In the quantum setting, we consider pairs of classical-quantum states with pure component states, and find that majorization is characterized by all ααα-zzz relative entropies for α<1α<1α<1. This demonstrates the first operational interpretation of these relative entropies.
This paper attempts to give an overview about sufficiency in the setting of quantum statistics. The basic concepts are treated in parallel to the measure theoretic case. It turns out that several classical examples and results have a noncommutative analogue. Some of the results are presented without proof (but with exact references) and the presentation is intended to be self-contained. The main examples discussed in the paper are related to the Weyl algebra and to the exponential family of states. The characterization of sufficiency in terms of quantum Fisher information is a new result.
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 $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 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.
Consistent families of locally defined probability distributions that do not admit a joint global distribution are known as contextual, with primary examples arising in quantum theory. In this paper, we study such families of distributions using the theory of simplicial distributions, and further develop the theory for possibilistic distributions defined over the Boolean semiring. We characterize possibilistic collapses of simplicial distributions geometrically using bundle scenarios. Using this characterization together with a new connectivity condition on the total space of a bundle scenario, we provide a criterion for detecting extremal simplicial distributions. In parallel, we develop an analogous theory for presheaves on simplicial complexes, describe possibilistic collapses of empirical models on them using event scenarios together with a categorical extremality condition, and relate the two frameworks via a comparison result. We provide examples of contextual simplicial distributions that arise from our criteria on scenarios of interest in quantum foundations, such as Bell scenarios and boundaries of standard simplices, the latter connecting to Vorob'ev's classical theorem on acyclic complexes.
We point out a correspondence between classical and quantum states, by showing that for every classical distribution over phase--space, one can construct a corresponding quantum state, such that in the classical limit of ℏ→0\hbar\to 0ℏ→0 the latter converges to the former with respect to all measurable quantities.