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Journal of Mathematical Physics

Not necessary but sufficient condition for the positivity of generalized Wigner functions

G. Mourgues, Marc Roy Feix, J. C. Andrieux, P. Bertrand

Published Oct 1, 1985• 10 citations
Publisher

Abstract

Many authors, both in quantum mechanics and signal theory, are concerned with a condition of positivity for the generalized Wigner functions. They give (and use) as necessary and sufficient the condition that the kernel be the characteristic function of another Wigner function. It is shown that the condition is not necessary, but only sufficient.

DOI
10.1063/1.526772

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Many authors, both in quantum mechanics and signal theory, are concerned with a condition of positivity for the generalized Wigner functions. They give (and use) as necessary and sufficient the condition that the kernel be the characteristic function of another Wigner function. It is shown that the condition is not necessary, but only sufficient.
We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schrödinger-cat states and higher-order cat states on a circle. More precisely, for Schrödinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.
We discuss two sets of conditions that are necessary and sufficient for a function defined on phase space to be a Wigner distribution function (WDF). The first set is well known and involves the function itself; the second set is less familiar and involves the function's ``symplectic'' Fourier transform. After explaining why these two sets are equivalent, we explore some properties and applications of the second one. Among other things, we show that that set includes the position-momentum uncertainty relations as a special case, and in doing so we give a new derivation of them. This derivation itself serves as the starting point for the discussion of a quantum-mechanical moment problem. It also enables us to exhibit a real-valued phase-space function that obeys the uncertainty relations but that is not a WDF.