  We show that measuring any two quantum states by a random POVM, under a
suitable definition of randomness, gives probability distributions having total
variation distance at least a universal constant times the Frobenius distance
between the two states, with high probability. This result gives us the first
sufficient condition and an information-theoretic solution for the following
quantum state distinction problem: given an a priori known ensemble of quantum
states, is there a single POVM that gives reasonably large total variation
distance between every pair of states from the ensemble? Our random POVM method
also gives us the first information-theoretic upper bound on the number of
copies required to solve the quantum state identification problem for general
ensembles, i.e., given some number of independent copies of a quantum state
from an a priori known ensemble, identify the state. The standard quantum
approach to solving the hidden subgroup problem (HSP) is a special case of the
state identification problem where the ensemble consists of so-called coset
states of candidate hidden subgroups. Combining Fourier sampling with our
random POVM result gives us single register algorithms using polynomially many
copies of the coset state that identify hidden subgroups having polynomially
bounded rank in every representation of the ambient group. These HSP algorithms
complement earlier results about the powerlessness of random Fourier sampling
when the ranks are exponentially large, which happens for example in the HSP
over the symmetric group. The drawback of random Fourier sampling based
algorithms is that they are not efficient because measuring in a random basis
is not. This leads us to the open question of efficiently implementable
pseudo-random measurement bases.
