  Discrimination between quantum states is a fundamental task in quantum
information theory. Given two arbitrary tensor-product quantum states (TPQS)
$\rho_{\pm} = \rho_{\pm}^{(1)} \otimes \cdots \otimes \rho_{\pm}^{(N)}$,
determining the joint $N$-system measurement to optimally distinguish between
the two states is a hard problem. Thus, there is great interest in identifying
local measurement schemes that are optimal or close-to-optimal. In this work,
we focus on distinguishing between two general TPQS. We begin by generalizing
previous work by Acin et al. (Phys. Rev. A 71, 032338) to show that a locally
greedy (LG) scheme using Bayesian updating can optimally distinguish between
two states that can be written as tensor products of arbitrary pure states.
Then, we show that even in the limit of large $N$ the same algorithm cannot
distinguish tensor products of mixed states with vanishing error probability.
This poor asymptotic behavior occurs because the Helstrom measurement becomes
trivial for sufficiently biased priors. Based on this, we introduce a modified
locally greedy (MLG) scheme with strictly better performance.
  In the second part of this work, we compare these simple local schemes with a
general dynamic programming (DP) approach that finds the optimal series of
local measurements to distinguish the two states. When the subsystems are
non-identical, we demonstrate that the ordering of the systems affects
performance and we extend the DP technique to determine the optimal ordering
adaptively. Finally, in contrast to the binary optimal collective measurement,
we show that adaptive protocols on sufficiently large (e.g., qutrit) subsystems
must contain non-binary measurements to be optimal. (The code that produced the
simulation results in this paper can be found at:
https://github.com/SarahBrandsen/AdaptiveStateDiscrimination)
