The compatibility dimension of quantum measurements

TL;DR

This paper introduces the compatibility dimension of quantum measurements, quantifies it using bounds, and analyzes specific cases like mutually unbiased bases to understand measurement compatibility in quantum systems.

Contribution

It defines the compatibility dimension for quantum measurements and provides bounds and detailed analysis for specific measurement sets, advancing understanding of measurement compatibility.

Findings

Bounds for the compatibility dimension are established.

Compatibility dimension is analyzed for two orthonormal bases.

Special case of mutually unbiased bases is examined.

Abstract

We introduce the notion of compatibility dimension for a set of quantum measurements: it is the largest dimension of a Hilbert space on which the given measurements are compatible. In the Schr\"odinger picture, this notion corresponds to testing compatibility with ensembles of quantum states supported on a subspace, using the incompatibility witnesses of Carmeli, Heinosaari, and Toigo. We provide several bounds for the compatibility dimension, using approximate quantum cloning or algebraic techniques inspired by quantum error correction. We analyze in detail the case of two orthonormal bases, and, in particular, that of mutually unbiased bases.