Quantum measurement incompatibility in subspaces

TL;DR

This paper introduces a framework for understanding measurement incompatibility in high-dimensional quantum systems by classifying how incompatibility behaves under subspace projections, with implications for quantum steering and measurement theory.

Contribution

It defines and characterizes three types of measurement incompatibility in subspaces, providing explicit examples and exploring their implications for quantum measurement concepts.

Findings

Incompatibility can be classified into three types in subspaces.

Joint measurability and coexistence are inequivalent in qubit systems.

Results impact tests of quantum steering.

Abstract

We consider the question of characterising the incompatibility of sets of high-dimensional quantum measurements. We introduce the concept of measurement incompatibility in subspaces. That is, starting from a set of measurements that is incompatible, one considers the set of measurements obtained by projection onto any strict subspace of fixed dimension. We identify three possible forms of incompatibility in subspaces: (i) incompressible incompatibility: measurements that become compatible in every subspace, (ii) fully compressible incompatibility: measurements that remain incompatible in every subspace, and (iii) partly compressible incompatibility: measurements that are compatible in some subspace and incompatible in another. For each class we discuss explicit examples. Finally, we present some applications of these ideas. First we show that joint measurability and coexistence are two inequivalent notions of incompatibility in the simplest case of qubit systems. Second we highlight the implications of our results for tests of quantum steering.