Density Matrices and Wigner Functions of Quasiclassical Quantum Systems

S. Chaturvedi, E. Ercolessi, G. Marmo +3 more

V. V. Dodonov, Vladimir I. Man’ko, D. L. Ossipov

As is well known, there are several formulations of quantum mechanics. One of them, closely connected with the Weyl representation of operators [ 1 ] and Wigner functions [ 2 ], use functions on the phase space to represent both observables and quantum states. For working in this representation it is necessary to replace operators by their Weyl symbols and density matrices by Wigner functions [ 3 , 4 ]. After fundamental works of Weyl, Wigner and Kirkwood [ 1 , 2 , 5 ] properties of the representation were studied and used to calculate the properties of various physical systems (see [ 6–23 ]), including systems with magnetic field [ 15–23 ].

A V Gorbachev

Helmut Hofmann

Abstract This chapter introduces density operators and matrices for many-body systems, and explains their reduction to one- and two-body densities. Starting from the Slater determinant, the latter are explicitly calculated for independent fermions. For such systems, the classical limit is discussed and formulas for partition function, free energy, and entropy are derived by introducing and employing the concept of Wigner transformations. The propagation of wave packets is described in one dimension, a case for which correspondence rules are derived for various combinations of quantum operators. Finally, formulas are given for the equilibrium distribution of the quantal oscillator and the associated Wigner function.

Leon W. Cohen

Pittenger, A O, Rubin, M H

A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension pn where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p2n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p=2) and our technique fails to establish the separability property for more than two qubits.