State and Measurement Design for Quantum Detection Over Quantum Channels

In quantum state discrimination, typically the design of measurement operators or probe state is formulated assuming the set of possible states is perfectly known, but this may yield designs which are sensitive to deviations in the realized set of states. For example, the channel through which a transmitted state is sent may not be deterministic, but instead characterized by a classical distribution over quantum channels. In this paper, we consider the design of measurement schemes and probe states for quantum detection over an uncertain quantum channel. We present stochastic gradient-based algorithms to maximize the expected performance over the channel distribution for various design objectives, including the detection probability and mutual information. Furthermore, we introduce a scheme that leverages the isometric extension of a quantum channel to measure the channel output in an enlarged Hilbert space such that the channels are more distinguishable, while simultaneously reducing the effective dimension and thereby reducing the optimization complexity. Finally, we apply the proposed algorithms to multicopy channel discrimination.