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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 set(assuming 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 the(near-)optimality of bipartite data hiding: The bias that can be achieved byLOCC in discriminating two orthogonal states of a d x d bipartite system isOmega(1/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.

Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding