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AIP conference proceedings

Non-stationary qubit states’ evolution in probability representation of quantum mechanics

Oleg Kulyashov, Vladimir I. Man’ko

Published Jan 1, 2021• 0 citations
Publisher

Abstract

The von Neuman equation for the qubit state density matrix is mapped onto the kinetic equations for probability distributions identified with the qubit states. In terms of these distributions known analytical solutions are analyzed: evolution of the probability distributions and of their Shannon and Tsallis entropies is demonstrated. The distributions are also used to describe (numerically) a special case of non-Hermitian evolution of a qubit.

DOI
10.1063/5.0055479

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The von Neuman equation for the qubit state density matrix is mapped onto the kinetic equations for probability distributions identified with the qubit states. In terms of these distributions known analytical solutions are analyzed: evolution of the probability distributions and of their Shannon and Tsallis entropies is demonstrated. The distributions are also used to describe (numerically) a special case of non-Hermitian evolution of a qubit.
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.
Review of probability representation of quantum mechanics where the system states are identified with fair probability distributions is presented. Examples of qubits and qutrits are given in detail. The explicit expressions of the qubit density matrices and qubit state vectors in terms of probabilities of dichotomic random variables are discussed. The wave function of a qubit state, and its phase which is not contained in the density matrix of the state, are expressed in terms of the dichotomic probabilities. The generalisation of the probability representation to the case of qudit state is considered. The Schrödinger–like equation for spectra of Hermitian and non Hermitian Hamiltonians is written in the new form of equation for the dichotomic probabilities.
Abstract A system of equations is obtained for a generalized qubit density matrix that depends on two moments in time. It determines the evolution of the quantum state of a qubit and transforms into the von Neumann equation for a qubit density matrix at coinciding times. The states of a harmonic oscillator in a probabilistic representation of quantum states are considered. Kinetic equations for the probabilities of projections of a spin +1/2 onto three perpendicular directions in space that determine the state of a qubit are obtained.
Abstract A review of the approach where the states of quantum systems are identified with fair probability distributions is presented. The quantizer–dequantizer operators used to construct the invertible map of the density operators onto the probability distributions are applied to obtain the kinetic equations for probability distributions identified with the quantum system states. For qubit states, the von Neumann evolution equation for the density operator is explicitly given in the form of kinetic equation for the probability distribution. Simplest tomographic probability distributions describing the states of multimode quantum oscillator are constructed.
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. Schrödinger 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.
The von Neumann evolution equation for density matrix and the Moyal equation for the Wigner function are mapped onto evolution equation for optical tomogram of quantum state. The connection with known evolution equation for symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for the optical tomograms. Classical Liouville equation for optical tomogram is obtained. Example of parametric oscillator is considered in detail.