Probability Representation of Quantum States

The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrodinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.

Automated summary

This review discusses the new formulation of conventional quantum mechanics where quantum states are regarded as probability distributions, and an invertible map of density operators and wave functions onto probability distributions is constructed. Examples of probability representations of qubits, harmonic oscillators, and free particles are studied in detail, and equations for the evolution of quantum systems are written in the form of linear classical-like equations for probability distributions. Relations to phase-space representation of quantum states and classical mechanics are also elucidated.