Optimal estimation and discrimination of excess noise in thermal and   amplifier channels
  We determine a fundamental upper bound on the performance of any adaptive protocol for discrimination or estimation of a channel which has an unknown parameter encoded in the state of its environment. Since our approach relies on the principle of data processing, the bound applies to a variety of discrimination measures, including quantum relative entropy, hypothesis testing relative entropy, R\'enyi relative entropy, fidelity, and quantum Fisher information. We apply the upper bound to thermal (amplifier) channels with a known transmissivity (gain) but unknown excess noise. In these cases, we find that the upper bounds are achievable for several discrimination measures of interest, and the method for doing so is non-adaptive, employing a highly squeezed two-mode vacuum state at the input of each channel use. Estimating the excess noise of a thermal channel is of principal interest for the security of quantum key distribution, in the setting where a fiber-optic cable has a known transmissivity but a tampering eavesdropper alters the excess noise on the channel, so that estimating the excess noise as precisely as possible is desirable. Finally, we outline a practical strategy which can be used to achieve these limits. 
Masahiro Takeoka and Mark M. Wilde
quant-ph