Probability Representation of Quantum Mechanics Where System States Are Identified with Probability Distributions

Abstract

The probability representation of quantum mechanics where the system states are identified with fair probability distributions is reviewed for systems with continuous variables (the example of the oscillator) and discrete variables (the example of the qubit). The relation for the evolution of the probability distributions which determine quantum states with the Feynman path integral is found. The time-dependent phase of the wave function is related to the time-dependent probability distribution which determines the density matrix. The formal classical-like random variables associated with quantum observables for qubit systems are considered, and the connection of the statistics of the quantum observables with the classical statistics of the random variables is discussed.

Quantum process in probability representation of quantum mechanics

“…where Ûmn ≡ Â † mn Û Âmn . From comparing (25) with (7) we see, that the sum mn Ûmn play here the role of the dequantizer after the operation is performed. Relationship (25), in turn, being written in terms of the scalar product of symbols becomes…”

“…For a further review of the symplectic tomography, we will employ the most convenient formalism involving the dequantizer and quantizer operators [4,7]. Let x = (X, µ, ν) be a vector containing three real variables.…”

“…To deal with functions associated with operators, so-called operator symbols, without losing the inherent to quantum mechanics noncommutativity, they are furnished with a star product formalism. All this machinery eventually brings the conventional quantum formalism into the domain of familiar functions and is well covered in the literature, see [4][5][6][7][8] and references therein.…”

“…At present, the probability representation of quantum mechanics has been already deeply elaborated and better suited to study quantum systems in a continuous-variable domain, but yet extended to deal with spin systems [7,9] and photon number states [10,11]. The scope of applications addressed using the symplectic tomography includes not only such particular problems as considering free quantum particles [8,12], particles in an electromagnetic field [13,14] or quantum oscillators [15,16], but also fundamental themes: open quantum systems [17,18], measurements [19], quantum information theory [20] and general aspects of quantum theory [21][22][23][24].…”

Editorial for the Special Issue “Selected Papers from the 16th International Conference on Squeezed States and Uncertainty Relations (ICSSUR 2019)”

“…; Man'ko, V.I. [2]. The result is based on the existence of invertible maps of the vectors and density operators acting in Hilbert spaces onto the probability distributions.…”