Operational criterion for Wigner function negativity

TL;DR

This paper presents an operational, experimentally accessible criterion to determine Wigner function negativity in quantum states using quantum non-demolition measurements, focusing on the role of coherent superpositions.

Contribution

It introduces a new criterion based on coherent-state superpositions that identifies Wigner function negativity, with proven necessity in specific quantum states.

Findings

The criterion is sufficient for Wigner function positivity.

For Schrödinger-cat states, the criterion is both necessary and sufficient.

In high-order cat states, the criterion applies in the large number limit.

Abstract

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: Schr\"{o}dinger-cat states and higher-order cat states on a circle. More precisely, for Schr\"{o}dinger-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.