  We analyze the distinguishability norm on the states of a multi-partite
system, defined by local measurements. Concretely, we show that the norm
associated to a tensor product of sufficiently symmetric measurements is
essentially equivalent to a multi-partite generalisation of the non-commutative
2-norm (aka Hilbert-Schmidt norm): in comparing the two, the constants of
domination depend only on the number of parties but not on the Hilbert spaces
dimensions.
  We discuss implications of this result on the corresponding norms for the
class of all measurements implementable by local operations and classical
communication (LOCC), and in particular on the leading order optimality of
multi-party data hiding schemes.
