  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.
