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Journal of Physics Conference Series

Nonstationary quantum systems and entanglement in the tomographic-probability representation

Vladimir I. Man’ko

Published Feb 1, 2008• 0 citations
Publisher

Abstract

Time-dependent integrals of motion for systems with both time-independent and time-dependent Hamiltonians are studied and expressed in terms of the evolution operator. The probability representation in which the system quantum states are described by tomographic probabilities (tomograms), is reviewed. Examples of systems of parametric oscillators and the entanglement phenomena induced by time-dependent coupling of the oscillators are discussed. Transition probabilities between the energy levels of parametric oscillators expressed in terms of special functions like Laguerre and Legendre polynomials are given in terms of tomographic probabilities.

DOI
10.1088/1742-6596/99/1/012013

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Time-dependent integrals of motion for systems with both time-independent and time-dependent Hamiltonians are studied and expressed in terms of the evolution operator. The probability representation in which the system quantum states are described by tomographic probabilities (tomograms), is reviewed. Examples of systems of parametric oscillators and the entanglement phenomena induced by time-dependent coupling of the oscillators are discussed. Transition probabilities between the energy levels of parametric oscillators expressed in terms of special functions like Laguerre and Legendre polynomials are given in terms of tomographic probabilities.
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We use the free evolution propagator to determine the quantum probability representation (i.e., the general expression of the tomogram) of any one-dimensional system described by a density state. The evolution operator for the considered quantum system is additionally used to establish the corresponding time dependent tomogram. Applications are given for a Gaussian wave packet, the quantum shutter related with the phenomenon of diffraction in time, the double quantum shutter, and a finite potential. A generalisation to describe NNN particle systems is also presented and, in particular, we find the tomogram associated to the 2 particle case occupying in general non-orthogonal states. In the latter case, for a bipartite quantum system, the entanglement properties are established by considering quantum information concepts such as the linear entropy.
We present a new formulation of conventional quantum mechanics, in which the notion of a quantum state is identified via a fair probability distribution of the position measured in a reference frame of the phase space with rotated axes. In this formulation, the quantum evolution equation as well as the equation for finding energy levels are expressed as linear equations for the probability distributions that determine the quantum states. We also give the integral transforms relating the probability distribution (called the tomographic-probability distribution or the state tomogram) to the density matrix and the Wigner function and discuss their connection with the Radon transform. Qudit states are considered and the invertible map of the state density operators onto the probability vectors is discussed. The tomographic entropies and entropic uncertainty relations are reviewed. We demonstrate the uncertainty relations for the position and momentum and the entropic uncertainty relations in the tomographic-probability representation, which is suitable for an experimental check of the uncertainty relations.
A short description of the notion of states of quantum systems in terms of conventional probability distribution function is presented. The notion and the structure of entangled probability distributions are clarified. The evolution of even and odd Schrödinger cat states of the inverted oscillator is obtained in the center-of-mass tomographic probability description of the two-mode oscillator. Evolution equations describing the time dependence of probability distributions identified with quantum system states are discussed. The connection with the Schrödinger equation and the von Neumann equation is clarified.
We review the probabilistic representation of quantum mechanics within which states are described by the probability distribution rather than by the wavefunction and density matrix. Uncertainty relations have been obtained in the form of integral inequalities containing measurable optical tomograms of quantum states. Formulas for the transition probabilities and purity parameter have been derived in terms of the tomographic probability distributions. Inequalities for Shannon and Rényi entropies associated with quantum tomograms have been obtained. A scheme of the star product of tomograms has been developed.
A review of the tomographic-probability representation of classical and quantum states is presented. The tomographic entropies and entropic uncertainty relations are discussed in connection with ambiguities in the interpretation of the state tomograms which are considered either as a set of the probability distributions of random variables depending on extra parameters or as a single joint probability distribution of these random variables and random parameters with specific properties of the marginals. Examples of optical tomograms of photon states, symplectic tomograms, and unitary spin tomograms of qudits are given. A new universal integral inequality for generic wave function is obtained on the base of tomographic entropic uncertainty relations.