Adaptive Procedures for Discrimination Between Arbitrary Tensor-Product\n Quantum States

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