Phase properties of operator valued measures in phase space

TL;DR

This paper explores the properties of operator valued measures in phase space, focusing on the Wigner phase operator, its eigenstates, and how to construct positive phase measurement operators using filtering techniques.

Contribution

It introduces a formalism for the Wigner phase distribution, constructs a positive operator valued measure (POVM) for phase, and links the Q-distribution to operational phase measurement schemes.

Findings

Eigenstates of the Wigner phase operator identified.

A positive phase operator formalism based on filtering is developed.

The Q phase POVM can be realized through interference at a beam-splitter.

Abstract

The Wigner Phase Operator (WPO) is identified as an operator valued measure (OVM) and its eigen states are obtained. An operator satisfying the canonical commutation relation with the Wigner phase operator is also constructed and this establishes a Wigner distribution based operator formalism for the Wigner Phase Distribution. The operator satisfying the canonical commutation relation with the Wigner Phase Operator valued measure (WP-OVM) is found to be not the usual number operator. We show a way to overcome the non-positivity problem of the WP-OVM by defining a positive OVM by means of a proper filter function, based on the view that phase measurements are coarse-grained in phase space, leading to the well known Q-distribution. The identification of Q phase operator as a POVM is in good agreement with the earlier observation regarding the relation between operational phase measurement schemes and the Q-distribution. The Q phase POVM can be dilated in the sense of Gelfand-Naimark, to an operational setting of interference at a beam-splitter with another coherent state - this results in a von Neumann projector with well-defined phase.