The complexity of compatible measurements

TL;DR

This paper investigates the structure and complexity of compatible quantum measurements, revealing bounds on their parent measurement outcomes and exploring the rich structure of extremal cases, highlighting a trade-off between memory and computational time.

Contribution

It establishes an upper bound on the outcomes of parent measurements for compatible sets, showing that complexity is generally limited and analyzing extremal cases for deeper understanding.

Findings

Existence of maximally complex parent measurements in simple scenarios

Upper bound on outcomes of parent measurements linear in the number of measurements

Rich structure of extremal compatible measurements uncovered

Abstract

Measurement incompatibility is one of the basic aspects of quantum theory. Here we study the structure of the set of compatible -- i.e. jointly measurable -- measurements. We are interested in whether or not there exist compatible measurements whose parent is maximally complex -- requiring a number of outcomes exponential in the number of measurements, and related questions. Although we show this to be the case in a number of simple scenarios, we show that generically it cannot happen, by proving an upper bound on the number of outcomes of a parent measurement that is linear in the number of compatible measurements. We discuss why this doesn't trivialise the problem of finding parent measurements, but rather shows that a trade-off between memory and time can be achieved. Finally, we also investigate the complexity of extremal compatible measurements in regimes where our bound is not tight, and uncover rich structure.