Adaptive Procedures for Discrimination Between Arbitrary Tensor-Product Quantum States

TL;DR

This paper develops and compares adaptive local measurement schemes for distinguishing tensor-product quantum states, demonstrating the limitations of greedy algorithms and proposing an optimal dynamic programming approach that adapts measurement order and type.

Contribution

It introduces a modified locally greedy scheme and an adaptive dynamic programming method for optimal quantum state discrimination, including strategies for non-identical subsystems and non-binary measurements.

Findings

Locally greedy scheme is optimal for pure states but fails for mixed states.

Dynamic programming can find optimal measurement sequences, considering subsystem ordering.

Adaptive protocols require non-binary measurements for large subsystems to achieve optimality.

Abstract

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)