Tilted phase-space measurements in the eight-port homodyne detection scheme

We show that the phase shift of $\frac{\pi }{2}$ 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 ${\mathbb{R}}^2$ 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.

Figure 1

Tilting of the phase space caused by a change in the phase shift.

Figure 2

Eight-port homodyne detector.

Figure 3

Distributions corresponding to $\theta =\frac{\pi }{2}$ (left) and $\theta =\frac{\pi }{4}$ (right) for a fixed initial state and generating operator.

Figure 4

Distributions corresponding to $\theta =\frac{\pi }{2}$ (left) and $\theta =\frac{\pi }{4}$ (right) for a fixed initial state and parameter field. The generating operator changes according to (6).