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Journal of Russian Laser Research

Probability Representation of Classical States

Olga V. Man’ko, Vladimir I. Man’ko, Oleg V. Pilyavets

Published Nov 1, 2005• 11 citations
Publisher
DOI
10.1007/s10946-005-0045-x

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•Aug 22, 2008•0 citations
Using the known possibility to associate the completely positive maps with density matrices and recent results on expressing the density matrices with sets of classical probability distributions of dichotomic random variables we construct the probability representation of the completely positive maps. In this representation, any completely positive map of qubit state density matrix is identified with the set of classical coin probability distributions. Examples of the maps of qubit states are studied in detail. The evolution equation of quantum states is written in the form of the classical-like kinetic equation for probability distributions identified with qubit state.
Several finite dimensional quasi-probability representations of \nquantum states have been proposed to study various problems in \nquantum information theory and quantum foundations. These \nrepresentations are often defined only on restricted dimensions and \ntheir physical significance in contexts such as drawing \nquantum-classical comparisons is limited by the non-uniqueness of \nthe particular representation. In this thesis it is shown how the mathematical \ntheory of frames provides a unified formalism which accommodates all \nknown quasi-probability representations of finite dimensional \nquantum systems. \n \nIt is also shown that any quasi-probability \nrepresentation is equivalent to a frame representation and it is \nproven that any such representation of quantum mechanics must exhibit \neither negativity or a deformed probability calculus. \n \nAlong the way, the connection between negativity and two other famous notions of non-classicality, namely contextuality and nonlocality, is clarified.