  Reinforcement learning with neural networks (RLNN) has recently demonstrated
great promise for many problems, including some problems in quantum information
theory. In this work, we apply RLNN to quantum hypothesis testing and determine
the optimal measurement strategy for distinguishing between multiple quantum
states $\{ \rho_{j} \}$ while minimizing the error probability. In the case
where the candidate states correspond to a quantum system with many qubit
subsystems, implementing the optimal measurement on the entire system is
experimentally infeasible.
  We use RLNN to find locally-adaptive measurement strategies that are
experimentally feasible, where only one quantum subsystem is measured in each
round. We provide numerical results which demonstrate that RLNN successfully
finds the optimal local approach, even for candidate states up to 20
subsystems. We additionally demonstrate that the RLNN strategy meets or exceeds
the success probability for a modified locally greedy approach in each random
trial.
  While the use of RLNN is highly successful for designing adaptive local
measurement strategies, in general a significant gap can exist between the
success probability of the optimal locally-adaptive measurement strategy and
the optimal collective measurement. We build on previous work to provide a set
of necessary and sufficient conditions for collective protocols to strictly
outperform locally adaptive protocols. We also provide a new example which, to
our knowledge, is the simplest known state set exhibiting a significant gap
between local and collective protocols. This result raises interesting new
questions about the gap between theoretically optimal measurement strategies
and practically implementable measurement strategies.
