Standard quantum mechanics with a probability distribution instead of a wave function

Vladimir I. Man’ko, Olga V. Man’ko

A new formulation of quantum mechanics (probability representation) is discussed. In this representation, a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function. An unambiguous relation (analog of Radon transformation) between the density operator and a tomogram is constructed both for continuous coordinates and for spin variables. A novel feature of a state, tomographic entropy, is considered, and its connection with von Neumann entropy is discussed. A one-to-one map of quantum observables (Hermitian operators) on positive probability distributions is found.

Peter Mittelstaedt

Lipo Wang, Robert F. O’Connell

Vladimir I. Man’ko, Oleg V. Pilyavets, Vadim Zborovskii

The probability representation of states in standard quantum mechanics where the quantum states are associated with fair probability distributions (instead of wave function or density matrix) is shortly commented and bibliography related to the probability representation is given.

Vladimir I. Man’ko, Guillaume Adenier, Andrei Yu. Khrennikov +3 more

Review of the probability representation of quantum mechanics, in which a fair probability distribution is used instead of the density matrix or wave function to describe quantum states is presented. Bell inequalities are formulated as properties of joint probability distributions describing the states of multipartite spin systems in the probability representation. Probability distributions depending on extra parameters are discussed in the connection with problems of classical and quantum statistical mechanics, where these distributions can be considered in the context of non‐Kolmogorov probability theory.

David Z. Albert