Showing papers by "Vladimir I. Man’ko published in 2021"
TL;DR: In this article, a review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented and the invertible map of density operators and wave functions onto the probability distributions describing 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.
Abstract: 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.
32 citations
TL;DR: In this article, the authors review the method of quantizers and dequantizers to construct an invertible map of the density operators onto functions including probability distributions and discuss in detail examples of qubit and qutrit states.
Abstract: We review the method of quantizers and dequantizers to construct an invertible map of the density operators onto functions including probability distributions and discuss in detail examples of qubit and qutrit states. The biphoton states existing in the process of parametric down-conversion are studied in the probability representation of quantum mechanics.
11 citations
TL;DR: In this article, the Wigner and tomographic representations of thermal Gibbs states for one and two-mode quantum systems described by a quadratic Hamiltonian are obtained by using the covariance matrix of the mentioned states.
Abstract: The Wigner and tomographic representations of thermal Gibbs states for one- and two-mode quantum systems described by a quadratic Hamiltonian are obtained. This is done by using the covariance matrix of the mentioned states. The area of the Wigner function and the width of the tomogram of quantum systems are proposed to define a temperature scale for this type of states. This proposal is then confirmed for the general one-dimensional case and for a system of two coupled harmonic oscillators. The use of these properties as measures for the temperature of quantum systems is mentioned.
5 citations
16 Jun 2021
TL;DR: In this paper, a review of probability representation of quantum mechanics where the system states are identified with fair probability distributions is presented and examples of qubits and qutrits are given in detail.
Abstract: 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 Schrodinger–like equation for spectra of Hermitian and non Hermitian Hamiltonians is written in the new form of equation for the dichotomic probabilities.
2 citations
TL;DR: In this article, the Schrodinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time was studied, and the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field.
Abstract: The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrodinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton’s law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.
1 citations
01 Jan 2021
1 citations