Fundamental Limits of Loss Sensing over Bosonic Channels

We consider the problem of estimating unknown loss η over
$n$
uses of single-mode lossy thermal noise bosonic channel under an average photon number constraint per mode. We prove that a product of
$n$
two-mode squeezed vacuum (TMSV) states achieves minimal quantum Cramér-Rao bound (QCRB) over Gaussian quantum states in this scenario, and characterize the optimal receiver structure. We show that TMSV minimizes QCRB over all quantum states in the limit of low input photon number. Finally, we compare the performance of our optimal receiver for TMSV to other receivers.