Tilted phase space measurements

TL;DR

This paper investigates how phase shifts affect phase space measurements in quantum optics, showing that different phase shifts require different mathematical representations to maintain covariance, thus enabling measurement determination for various parameters.

Contribution

It introduces a new approach to handle arbitrary phase shifts in phase space measurements by constructing alternative projective representations.

Findings

The phase shift of π/2 is essential for covariance in high-amplitude limits.

Explicit construction of a different representation for arbitrary phase shifts.

Determination of the measured observable for any phase shift and parameter field.

Abstract

We show that the phase shift of {\pi}/2 is crucial for the phase space translation covariance of the measured high-amplitude limit observable in eight-port homodyne detection. However, for an arbitrary phase shift {\theta} we construct explicitly a different nonequivalent projective representation of R$^2$ such that the observable is covariant with respect to this representation. As a result we are able to determine the measured observable for an arbitrary parameter field and phase shift. Geometrically the change in the phase shift corresponds to the tilting of one axis in the phase space of the system.