# Bounding the quantum limits of precision for phase estimation with loss and thermal noise

## TL;DR

This paper derives fundamental quantum limits on phase estimation accuracy in lossy, noisy optical systems, providing bounds on the mean-squared error achievable with optimal quantum probes.

## Contribution

It introduces an upper bound on quantum Fisher information for phase estimation under loss and thermal noise, advancing understanding of quantum sensing limits.

## Key findings

- Upper bound on quantum Fisher information as a function of system parameters
- Lower bound on mean-squared error for unbiased estimators
- Implications for fundamental limits of covert sensing

## Abstract

We consider the problem of estimating an unknown but constant carrier phase modulation $\theta$ using a general -- possibly entangled -- $n$-mode optical probe through $n$ independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\theta$ as a function of $n$, the mean and variance of the total number of photons $N_{\rm S}$ in the $n$-mode probe, the transmissivity $\eta$ and mean thermal photon number per mode ${\bar n}_{\rm B}$ of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-squared error (MSE) of an unbiased estimator $\tilde{\theta}$ of $\theta$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing, and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field.