Multiple-copy state discrimination: Thinking globally, acting locally

TL;DR

This paper compares various quantum state discrimination schemes for multiple copies, showing that adaptive local measurements outperform others for finite copies, but all schemes converge in performance asymptotically.

Contribution

It provides a comprehensive analysis of local versus global measurement schemes, highlighting the conditions where adaptive local measurements offer advantages.

Findings

Adaptive measurements outperform fixed schemes for finite copies.

All schemes converge in error probability as number of copies approaches infinity.

Naive fixed local measurements are nearly optimal for states with more than 2% mixture.

Abstract

We theoretically investigate schemes to discriminate between two nonorthogonal quantum states given multiple copies. We consider a number of state discrimination schemes as applied to nonorthogonal, mixed states of a qubit. In particular, we examine the difference that local and global optimization of local measurements makes to the probability of obtaining an erroneous result, in the regime of finite numbers of copies $N$, and in the asymptotic limit as $N \rightarrow \infty$. Five schemes are considered: optimal collective measurements over all copies, locally optimal local measurements in a fixed single-qubit measurement basis, globally optimal fixed local measurements, locally optimal adaptive local measurements, and globally optimal adaptive local measurements. Here, adaptive measurements are those for which the measurement basis can depend on prior measurement results. For each of these measurement schemes we determine the probability of error (for finite $N$) and scaling of this error in the asymptotic limit. In the asymptotic limit, adaptive schemes have no advantage over the optimal fixed local scheme, and except for states with less than 2% mixture, the most naive scheme (locally optimal fixed local measurements) is as good as any noncollective scheme. For finite $N$, however, the most sophisticated local scheme (globally optimal adaptive local measurements) is better than any other noncollective scheme, for any degree of mixture.