# Distinguishing $n$ pure states in an $n$ dimensional Hilbert space

Suppose we have $$n$$ pure states in an $$n$$ dimensional Hilbert space, and we would like to distinguish them using POVM or PVM. We get any one of the pure states with equal probability, and we may set the metric to be the average or the worst-case success probability.

Are there examples where POVM can do better than projective measurement (or a known proof it does not)?

• In finite dimensions, anything a POVM can do, there is a PVM with the same outcome probabilities via a Naimark dilation. Jun 20, 2022 at 17:19
• But this PVM will be in a larger space? Can we still have a PVM that is as good as the POVM in this $n$ dimensional space?
– Stan
Jun 20, 2022 at 17:43
• You are correct, the dilation gives a projection in a larger space. Jun 20, 2022 at 19:46
• Not sure what you mean by "we would like to distinguish them" Jun 21, 2022 at 15:21
• @Dani007 probably to find the POVM that optimally discriminates between the elements of the ensemble. In other words, the POVM $\mu$ that maximises the quantity $\sum_a \langle\mu_a,\rho_a\rangle$ with $\rho_a$ elements of the ensemble (assumed to be equiprobable). Or in this case, whether the optimal such POVM is (or at least can be chosen to be) a PVM, when there's $n$ different $\rho_a$ each of which is pure.
– glS
Jun 21, 2022 at 18:01