Operator formalism for the Wigner phase distribution

TL;DR

This paper develops a hermitian phase operator for the Wigner phase distribution, establishing its completeness, relation to the Pegg-Barnett formalism, and consistency with expected phase properties of quantum states.

Contribution

It introduces a hermitian operator for the Wigner phase, demonstrating its completeness and connection to existing phase formalisms, enhancing understanding of quantum phase measurements.

Findings

The operator is complete and associated with non-orthogonal states.

It satisfies a weak equivalence with the Pegg-Barnett phase operator.

Results align with the uniform phase distribution of Fock states.

Abstract

The probability distribution for finding a state of the radiation field in a particular phase is described by a multitude of theoretical formalisms; the phase-sensitivity of the Wigner quasi-probability distribution being one of them. We construct a hermitian phase operator for this Wigner phase. We show that this operator is complete and also elucidate a set of complete but non-orthogonal states that seems to be naturally associated with such an operator. Further we show that our operator satisfies a weak equivalence relation with the Pegg-Barnett operator, thus showing that the essential phase information furnished by both formalisms are the same. It is also shown that this operator gives results which are in correct agreement with the expected uniform phase distribution of a Fock state.