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Journal of Physics A Mathematical and General

Tomographic quantum measures for many degrees of freedom and the central limit theorem

Grigori Gennadievich Amosov, Vladimir I. Man’ko

Published Feb 24, 2005• 5 citations
Publisher

Abstract

A tomographic quantum measure for a multimode system is introduced. Symplectic tomograms describing quantum states of the system with many degrees of freedom are shown to be equal to partial derivatives of the von Neumann probability distribution functions of homodyne random variables. The central limit theorem known in quantum probability theory is applied to describe properties of the symplectic quantum measures introduced. An example of the centre-of-mass homodyne quadrature is studied in the context of the central limit theorem.

DOI
10.1088/0305-4470/38/10/008

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A tomographic quantum measure for a multimode system is introduced. Symplectic tomograms describing quantum states of the system with many degrees of freedom are shown to be equal to partial derivatives of the von Neumann probability distribution functions of homodyne random variables. The central limit theorem known in quantum probability theory is applied to describe properties of the symplectic quantum measures introduced. An example of the centre-of-mass homodyne quadrature is studied in the context of the central limit theorem.
We investigate the dependence of the center-of-mass tomogram of a system with many degrees of freedom N on Planck's constant . It is shown that the use of the central limit theorem at N→+∞ in the quantum case can be employed for checking the condition of its validity in the famous Lyapunov form. In this case, the resulting distribution is Gaussian if the initial distribution is a product of either independent excited states of a quantum oscillator or even and odd coherent states. Then, at Planck's constant → 0, one gets the δ-function associated with the distribution with the maximum at zero and with the probability equal to one.
The tomographic map of quantum state of a system with several degrees of freedom which depends on one random variable, analogous to center of mass considered in rotated and scaled reference frame in the phase space, is constructed. Time evolution equation of the tomogram for this map is given in explicit form. Properties of the map like the transition probabilities between the different states and relation to the star product formalism are elucidated. Example of multimode oscillator is considered in details. Symmetry properties in respect to identical particles permutations are discussed in the framework of proposed tomography scheme.
We investigate the dependence of the center-of-mass tomogram of a system with many degrees of freedom NNN on the Planck constant $\hbar $. It is shown that to use the central limit theorem under taking the limit $N\to +\infty $ one should keep the energy of the system to be constant. In the case, the resulting distribution is Gaussian if the initial distribution is a product of independent excited states of a quantum oscillator or even and odd coherent states either. Then, if one turns the Planck constant ℏ→0\hbar \to 0ℏ→0 we get δδδ-function associated with the distribution concentrated in zero with the probability equal to one.
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.
Partial symplectic conditional and joint probability representations of quantum mechanics are considered. The correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators are derived. Calculations were made by use of general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations. Taking the Gaussian functions as the distributions of the tomographic parameters the examples of joint probability representations were considered. Evolution equations and stationary states equations for partial symplectic conditional and joint probability distributions are obtained.