Quantum Detection Over Quantum Channels With Uncertainty

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 under two design scenarios, joint design and two-stage design. We consider various design objectives, including the detection probability and mutual information, with the latter leading to a hybrid scheme consisting of von Neumann measurement and classical hypothesis test. Furthermore, we introduce a channel discrimination scheme which leverages the isometric extension of a quantum channel which increases the channel distinguishability, while simultaneously reducing the effective dimension and thereby reducing the optimization complexity. In addition, we apply amortized optimization to train a recurrent neural network in order to improve the convergence speed of the proposed algorithms. Finally, we apply the proposed algorithms to multi-copy channel discrimination as well as a novel joint channel-state discrimination scenario.