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We determine a fundamental upper bound on the performance of any adaptiveprotocol for discrimination or estimation of a channel which has an unknownparameter encoded in the state of its environment. Since our approach relies onthe principle of data processing, the bound applies to a variety ofdiscrimination measures, including quantum relative entropy, hypothesis testingrelative entropy, R\'enyi relative entropy, fidelity, and quantum Fisherinformation. We apply the upper bound to thermal (amplifier) channels with aknown transmissivity (gain) but unknown excess noise. In these cases, we findthat the upper bounds are achievable for several discrimination measures ofinterest, and the method for doing so is non-adaptive, employing a highlysqueezed two-mode vacuum state at the input of each channel use. Estimating theexcess noise of a thermal channel is of principal interest for the security ofquantum key distribution, in the setting where a fiber-optic cable has a knowntransmissivity but a tampering eavesdropper alters the excess noise on thechannel, so that estimating the excess noise as precisely as possible isdesirable. Finally, we outline a practical strategy which can be used toachieve these limits.

Optimal estimation and discrimination of excess noise in thermal and amplifier channels