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arXiv:1012.2650

A quantum version of randomization criterion

Keiji Matsumoto

Published Dec 13, 2010• 14 citations
arXivHTMLPDF

Abstract

In classical statistical decision theory, comparison of experiments plays very important role. Especially, so-called randomization criteria is most important. In this paper, we establish two kinds of quantum analogue these concepts, and apply to some examples.
DOI
10.48550/arxiv.1012.2650

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The classical randomization criterion is an important result of statistical decision theory. Recently, a quantum analogue has been proposed, giving equivalent conditions for two sets of quantum states, ensuring existence of a quantum channel mapping one set close to the other, in L1L_1L1​-distance. In the present paper, we extend these concepts in several ways. First, sets of states are replaced by channels and randomization is performed by either post- or pre-composition with another channel. The L1L_1L1​-distance is replaced by the diamond norm. Secondly, the maps are not required to be completely positive, but positivity is given by an admissible family of convex cones. It is shown that the randomization theorems, generalizing both quantum and classical randomization criteria, can be proved in the framework of base section norms, including the diamond norm and its dual. The theory of such norms is developed in the Appendix.
A two-dimensional scattering potential represents the quantum extension of a diffractive lattice: a Dirac delta function with a modulated permeability along the y -axis. This model does not have an explicit classical analogue and quantum effects such as tunneling and diffraction play an important role. An analytical solution for the one-harmonic case is found. For the general case of an arbitrary number of harmonics a simple criterion is derived for the range of parameters where quantum chaos is permitted (but does not necessarily occur). The statistical properties of the S -matrix for the given model have been investigated. The deviations from the usual predictions for irregular scattering in the random matrix theory (RMT) framework have been found and are discussed.
Abstract This paper argues that von Neumann entropy plays two conceptually distinct roles in quantum theory. When applied to global mixed states, it expresses the quantum analogue of the informational entropy. But when applied to reduced states of entangled systems, it measures objective physical correlations. We propose that this second usage realizes a new informational kind , which we call relational information . Unlike information in communication theory, it reflects structural interdependence between systems, not probability about a certain outcome. We further suggest that this informational kind has ontological significance: relational information is a physically instantiated feature of entangled systems, and not merely an agent-relative concept.