Find an analytical state representation for phase-subsystem evaluation

Determine an alternative representation of the quantum state that makes the expectation value of the field-theoretic phase-subsystem operator \(\tilde{\Phi}_{s,V}(\phi|\mathbf{r})\) analytically tractable, thereby avoiding reliance on numerical integration for computing the local phase-subsystem distribution \(\rho_{s,V}(\phi|\mathbf{r})\).

Background

The supplementary analysis defines local phase subsystems for a point detector through the operator $\hat{\Phi}_{s,V}(\phi|\mathbf{r})$, whose expectation value gives the probability of observing phase $\phi$ at position $\mathbf{r}$. For a single-mode state represented by a Glauber $P$-function, this expectation value reduces to an integral involving the function $\theta(x)$, defined by an infinite double series.

The paper states that $\theta(x)$ has no simple analytical form and that an alternative representation of the state that would make the phase-subsystem expectation value more manageable is not known. Consequently, the authors rely on numerical integration, supplemented by numerical approximations to $\theta(x)$. Finding such a representation would provide a more analytical and potentially more efficient route to evaluating local phase-subsystem distributions.