  Every sufficiently rich set of measurements on a fixed quantum system defines
a statistical norm on the states of that system via the optimal bias that can
be achieved in distinguishing the states using measurements from that set
(assuming equal priors). The Holevo-Helstrom theorem says that for the set of
all measurements this norm is the trace norm. For finite dimension any norm is
lower and upper bounded by constant (though dimension dependent) multiples of
the trace norm, so we set ourselves the task of computing or bounding the best
possible "constants of domination" for the norms corresponding to various
restricted sets of measurements, thereby determining the worst case and best
case 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-designs
and 4-designs respectively. Here we find asymptotically tight bounds for the
constants of domination.
  Furthermore, we analyse the multipartite setting with any LOCC measurement
allowed. In the case of two parties, we show that the lower domination constant
is the same as that of a tensor product of local uniformly random POVMs (up to
a constant). This answers in the affirmative an open question about the
(near-)optimality of bipartite data hiding: The bias that can be achieved by
LOCC in discriminating two orthogonal states of a d x d bipartite system is
Omega(1/d), which is known to be tight. Finally, we use our analysis to derive
certainty relations (in the sense of Sanchez-Ruiz) for any such measurements
and to lower bound the locally accessible information for bipartite systems.
