Locally restricted measurements on a multipartite quantum system: data hiding is generic

TL;DR

This paper investigates the effectiveness of various locally restricted quantum measurements in distinguishing multipartite quantum states, revealing that some measurement classes perform nearly as well as unrestricted ones, while others show significant limitations.

Contribution

It provides a detailed analysis of the performance of different classes of locally restricted measurements, highlighting their strengths and limitations in high-dimensional quantum state discrimination.

Findings

PPT measurements perform almost as well as all measurements.

Local and separable measurements have substantial performance loss.

Approximately exp(d^2) POVMs are needed for near-perfect state discrimination.

Abstract

We study the distinguishability norms associated to families of locally restricted POVMs on multipartite systems. These norms (introduced by Matthews, Wehner and Winter) quantify how quantum measurements, subject to locality constraints, perform in the task of discriminating two multipartite quantum states. We mainly address the following question regarding the behaviour of these distinguishability norms in the high-dimensional regime: On a bipartite space, what are the relative strengths of standard classes of locally restricted measurements? We show that the class of PPT measurements typically performs almost as well as the class of all measurements whereas restricting to local measurements and classical communication, or even just to separable measurements, implies a substantial loss. We also provide examples of state pairs which can be perfectly distinguished by local measurements if (one-way) classical communication is allowed between the parties, but very poorly without it. Finally, we study how many POVMs are needed to distinguish almost perfectly any pair of states on $\mathbf{C}^d$, showing that the answer is $\exp(\Theta(d^2))$.