Attainability of quantum state discrimination bounds with collective measurements on finite copies

One of the fundamental tenets of quantum mechanics is that nonorthogonal states cannot be distinguished perfectly. When distinguishing multiple copies of a mixed quantum state, a collective measurement, which generates entanglement between the different copies of the unknown state, can achieve a lower error probability than nonentangling measurements. The error probability that can be attained using a collective measurement on a finite number of copies of the unknown state is given by the Helstrom bound. In the limit where we can perform a collective measurement on asymptotically many copies of the quantum state, the quantum Chernoff bound gives the attainable error probability. It is natural to ask, then, what strategies can be employed to reach these two bounds---is entanglement across all available modes always necessary or can other, experimentally more simple measurements saturate these bounds? In this work we address this question. We find analytic expressions for the Helstrom bound for arbitrarily many copies of the unknown state in simple qubit examples. Using these analytic expressions, we investigate whether the quantum Chernoff bound can be saturated by repeatedly implementing the $M$-copy Helstrom measurement. We also investigate the necessary conditions to saturate the $M$-copy Helstrom bound. It is known that a collective measurement on all $M$ copies of the unknown state is always sufficient to saturate the $M$-copy Helstrom bound. However, general conditions for when such a measurement is necessary to saturate the Helstrom bound remain unknown. We investigate specific measurement strategies which involve entangling operations on fewer than all $M$ copies of the unknown state. For many regimes we find that a collective measurement on all $M$ copies of the unknown state is necessary to saturate the $M$-copy Helstrom bound.