# Fundamental limits of quantum-secure covert optical sensing

## TL;DR

This paper establishes a fundamental limit on active quantum optical sensing, showing that the mean squared error scales as 1/√n for covert phase estimation, and this limit is achievable with practical laser and heterodyne detection.

## Contribution

It derives a square root law for covert optical sensing and characterizes the fundamental limit, demonstrating its achievability with realistic quantum measurement techniques.

## Key findings

- Mean squared error scales as 1/√n for covert sensing.
- Achievability of the limit using laser illumination and heterodyne detection.
- Any attempt to improve accuracy beyond this limit risks detection by an adversary.

## Abstract

We present a square root law for active sensing of phase $\theta$ of a single pixel using optical probes that pass through a single-mode lossy thermal-noise bosonic channel. Specifically, we show that, when the sensor uses an $n$-mode covert optical probe, the mean squared error (MSE) of the resulting estimator $\hat{\theta}_n$ scales as $\langle (\theta-\hat{\theta}_n)^2\rangle=\mathcal{O}(1/\sqrt{n})$; improving the scaling necessarily leads to detection by the adversary with high probability. We fully characterize this limit and show that it is achievable using laser light illumination and a heterodyne receiver, even when the adversary captures every photon that does not return to the sensor and performs arbitrarily complex measurement as permitted by the laws of quantum mechanics.