Naimark dilations of qubit POVMs and joint measurements

TL;DR

This paper introduces a novel method using Naimark dilations to determine measurement compatibility in qubit systems, providing a comprehensive analytical framework that complements existing numerical and analytical approaches.

Contribution

The authors develop a general approach based on minimal Naimark dilations to characterize measurement compatibility, including for binary, ternary, and continuous qubit measurements.

Findings

Characterizes compatible measurements via block-diagonal representations in Naimark dilations.

Retrieves the Busch criterion for binary qubit measurements.

Applies the method to special cases of ternary and continuous qubit measurements.

Abstract

Measurement incompatibility is one of the cornerstones of quantum theory. This phenomenon appears in many forms, of which the concept of non-joint measurability has received considerable attention in the recent years. In order to characterise this non-classical phenomenon, various analytical and numerical methods have been developed. The analytical approaches have mostly concentrated on the qubit case, as well as to scenarios involving sets of measurements with symmetries, such as position and momentum or sets of mutually unbiased bases. The numerical methods can, in principle, decide any finite-dimensional and discrete joint measurability problem, but they naturally have practical limitations in terms of computational power. These methods exclusively start from a given set of measurements and ask whether the set possesses incompatibility. Here, we take a complementary approach by asking which measurements are compatible with a given measurement. It turns out, that this question can be answered in full generality through a minimal Naimark dilation of the given measurement: the set of interest is exactly those measurements that have a block-diagonal representation in such dilation. We demonstrate the use of the technique through various qubit examples, leading to an alternative characterisation of all compatible pairs of binary qubit measurements, which retrieves the celebrated Busch criterion. We further apply the technique to special examples of trinary and continuous qubit measurements.