# Optimal lossy quantum interferometry in phase space

## TL;DR

This paper investigates the phase space representation of optimal phase measurement in lossy interferometers, deriving bounds on precision and analyzing the effects of photon loss on measurement sensitivity.

## Contribution

It introduces a phase space approach to quantify the impact of photon loss on quantum interferometry and derives bounds on measurement precision in lossy conditions.

## Key findings

- Identifies features of the spin Wigner function that enable sub-shot noise precision.
- Derives the asymptotic form of the photon loss kernel in phase space.
- Provides a lower bound on quantum Fisher information based on Wigner functions.

## Abstract

We analyse the phase space representation of the optimal measurement of a phase shift in an interferometer with equal photon loss in both its arms. In the local phase estimation scenario with a fixed number of photons, we identify features of the spin Wigner function that warrant sub-shot noise precision, and discuss their sensitivity to losses. We derive the asymptotic form of an integral kernel describing the process of photon loss in the phase space in the limit of large photon numbers. The analytic form of this kernel allows one to assess the ultimate precision limit for a lossy interferometer. We also provide a general lower bound on the quantum Fisher information in terms of spin Wigner functions.