Local and global statistical distances are equivalent on pure states

TL;DR

This paper proves that for pure quantum states, the statistical distance remains the same whether measured globally or locally with classical communication, simplifying quantum state discrimination.

Contribution

It demonstrates the equivalence of global and local statistical distances for pure states under LOCC restrictions, revealing a fundamental property of quantum state geometry.

Findings

Statistical distance equals the Hilbert space angle for pure states.

Local operations with classical communication suffice for optimal state discrimination.

Global and local measurement strategies are equivalent in this context.

Abstract

The statistical distance between pure quantum states is obtained by finding a measurement that is optimal in a sense defined by Wootters. As such, one may expect that the statistical distance will turn out to be different if the set of possible measurements is restricted in some way. It nonetheless turns out that if the restriction is to local operations and classical communication (LOCC) on any multipartite system, then the statistical distance is the same as it is without restriction, being equal to the angle between the states in Hilbert space.