Mixed states with positive Wigner functions

authored by

T. Bröcker, R. F. Werner

Abstract

According to a result of Hudson, the Wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent. The characterization of mixed states with positive Wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters (i.e. functions with given Wigner spectrum in the sense of Narcowich). If a state is a convex combination of coherent states, it has the property that the Wigner spectrum contains the unit interval, which in turn implies that the Wigner function is positive. We show by explicit examples that the converses of both implications are false. The examples are taken from a low dimensional section of the state space, in which all Wigner spectra can be computed. In this set we also find counterexamples to a conjecture by Narcowich concerning the Wigner spectrum of products, as well as a state whose Wigner spectrum is a convergent sequence of discrete points.

Organisation(s)

Institute of Theoretical Physics

Type

Article

Journal

J. Math. Phys.

Volume

36

Pages

62-75

No. of pages

14

Publication date

1995

Publication status

Published

Peer reviewed

Yes

Electronic version(s)

https://doi.org/10.1063/1.531326 (Access: Unknown)