Random measurement bases, quantum state distinction and applications to the hidden subgroup problem

We show that measuring any two low rank quantum states in a random orthonormal basis gives, with high probability, two probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states. This implies that for any finite ensemble of quantum states there is a single POVM that distinguishes between every pair of states from the ensemble by at least a constant times their Frobenius distance; in fact, with high probability a random POVM, under a suitable definition of randomness, suffices. There are examples of ensembles with constant pairwise trace distance where a single POVM cannot distinguish pairs of states by much better than their Frobenius distance, including the important ensemble of coset states of hidden subgroups of the symmetric group (Moore at al., 2005). We next consider the random Fourier method for the hidden subgroup problem (HSP) which consists of Fourier sampling the coset state of the hidden subgroup using random orthonormal bases for the group representations. In cases where every representation of the group has polynomially bounded rank when averaged over the hidden subgroup, the random Fourier method gives a POVM for the HSP operating on one coset state at a time and using totally a polynomial number of coset states. In particular, we get such POVMs whenever the group and the hidden subgroup form a Gel´fand pair, e.g., Abelian, dihedral and Heisenberg groups. This gives a positive counterpart to earlier negative results about random Fourier sampling when the above rank is exponentially large (Grigni et al., 2004), which happens for example in the HSP in the symmetric group. The drawback of random POVMs is that they are not efficient to implement, since measuring in a random basis takes exponential time as can be seen by a counting argument. This leads us to the open question of efficiently implementable pseudorandom measurement bases