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Abstract: Every sufficiently rich set of measurements on a fixed quantum system definesa statistical norm on the states of that system via the optimal bias that canbe achieved in distinguishing the states using measurements from that setassuming equal priors. The Holevo-Helstrom theorem says that for the set ofall measurements this norm is the trace norm. For finite dimension any norm islower and upper bounded by constant though dimension dependent multiples ofthe trace norm, so we set ourselves the task of computing or bounding the bestpossible -constants of domination- for the norms corresponding to variousrestricted sets of measurements, thereby determining the worst case and bestcase performance of these sets relative to the set of all measurements.We look at the case where the allowed set consists of a single measurement,namely the uniformly random continuous POVM and its approximations by 2-designsand 4-designs respectively. Here we find asymptotically tight bounds for theconstants of domination.Furthermore, we analyse the multipartite setting with any LOCC measurementallowed. In the case of two parties, we show that the lower domination constantis the same as that of a tensor product of local uniformly random POVMs up toa constant. This answers in the affirmative an open question about thenear-optimality of bipartite data hiding: The bias that can be achieved byLOCC in discriminating two orthogonal states of a d x d bipartite system isOmega1-d, which is known to be tight. Finally, we use our analysis to derivecertainty relations in the sense of Sanchez-Ruiz for any such measurementsand to lower bound the locally accessible information for bipartite systems.



Author: William Matthews, Stephanie Wehner, Andreas Winter

Source: https://arxiv.org/



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