Online identification of symmetric pure states

TL;DR

This paper develops and analyzes online measurement strategies for discriminating symmetric pure quantum states with zero error, demonstrating their optimality and robustness in various scenarios, and identifying limitations for complex overlaps.

Contribution

It introduces optimal online schemes for zero-error discrimination of symmetric states, extending previous binary results to multiple states and analyzing their performance limits.

Findings

Online schemes are optimal for positive overlaps and odd n with negative overlaps.

Online strategies require minimal classical memory and are robust against particle losses.

For complex overlaps, online schemes likely do not achieve global optimality.

Abstract

We consider online strategies for discriminating between symmetric pure states with zero error when $n$ copies of the states are provided. Optimized online strategies involve local, possibly adaptive measurements on each copy and are optimal at each step, which makes them robust in front of particle losses or an abrupt termination of the discrimination process. We first review previous results on binary minimum and zero error discrimination with local measurements that achieve the maximum success probability set by optimizing over global measurements, highlighting their online features. We then extend these results to the case of zero error identification of three symmetric states with constant overlap. We provide optimal online schemes that attain global performance for any $n$ if the state overlaps are positive, and for odd $n$ if overlaps have a negative value. For arbitrary complex overlaps, we show compelling evidence that online schemes fail to reach optimal global performance. The online schemes that we describe only require to store the last outcome obtained in a classical memory, and adaptiveness of the measurements reduce to at most two changes, regardless of the value of $n$.