Single-shot comparison of random quantum channels and measurements

We analyze the task of deciding whether two unknown quantum devices are identical or independent, for arbitrary quantum channels and generalized measurements (POVMs) drawn at random from Haar-Stinespring ensembles, in the single-shot regime in which each device is used exactly once. Our contribution is twofold. First, in the symmetric setting, in which both types of error are treated on an equal footing, we derive a closed-form expression for the optimal success probability and show that it is achieved by a universal, parameter-free strategy: an antisymmetric two-copy input state followed by a projection onto the symmetric and antisymmetric subspaces of the outputs, with no additional ancillary systems required. Second, in the asymmetric (hypothesis-testing) setting we reduce the problem to a simple linear program and derive explicit type-I/type-II error trade-off curves for deterministic fixed-input protocols, for which the optimal input depends on the allowed type-I error; remarkably, classical randomization—equivalently, an untouched side register correlated with the input choice—can strictly improve the asymmetric performance by convexifying the deterministic trade-off, in sharp contrast to the symmetric setting. Finally, we show that the type-I error can be made to vanish without trivializing the test only in the environment-free case, corresponding to isometric channels, and for von Neumann measurements; for any genuinely noisy dynamics both types of error remain strictly positive in every nontrivial comparison scheme.