Coherent Protocols for Distinguishing Noisy Quantum Channels

A simple instance of quantum hypothesis testing involves identifying from which of two distributions a single-qubit rotation is drawn. As a twist on this well-known problem, suppose multiple rotations can be cascaded together coherently, rather than requiring that the result of each rotation be measured independently. Assuming independent measurements, an optimal strategy is entirely defined by the Helstrom bound for distinguishing states. Given the possible coherent protocol, what is the optimal error-minimizing procedure for solving this inference problem? Distinguishing channels as opposed to states offers opportunities for coherent protocols to improve over independent measurements. We show that if the coherent protocol is used, then for narrow distributions the number of queries required to achieve a particular inference confidence is reduced compared to simple Helstrom measurements. The constructive protocol employs quantum signal processing, performing polynomial transforms on the underlying distributions in a way that applies flexibly to many channel inference scenarios. This work expands the concept of quantum advantage to the statistical inference problem of quantum channel property testing.