Local sampling of the SU(1,1) Wigner function

TL;DR

This paper introduces a novel SU(1,1) Wigner function, derived using parity operator properties, and proposes an optical sampling scheme for direct measurement, enhancing the representation of SU(1,1) quantum states.

Contribution

It develops a consistent definition of the SU(1,1) Wigner function and proposes an optical method for its direct sampling, filling a gap in phase-space quantum state analysis.

Findings

Derived a bona fide SU(1,1) Wigner function using parity operator properties.

Proposed an optical scheme with a squeezer and photon detectors for direct sampling.

Provides a framework for accurate representation of SU(1,1) states.

Abstract

Despite the indisputable merits of the Wigner phase-space formulation, it has not been widely explored for systems with SU(1,1) symmetry, as a simple operational definition of the Wigner function has proved elusive in this case. We capitalize on the unique properties of the parity operator, to derive in a consistent way a \emph{bona fide} SU(1,1) Wigner function that faithfully parallels the structure of its continuous-variable counterpart. We propose an optical scheme, involving a squeezer and photon-number-resolving detectors, that allows for direct point-by-point sampling of that Wigner function. This provides an adequate framework to represent SU(1,1) states satisfactorily.