On compatibility of binary qubit measurements

TL;DR

This paper provides a complete geometric characterization of joint measurability for finite sets of unbiased binary qubit measurements using Boolean hypercube functions and Fourier transforms, with implications for quantum steering and measurement incompatibility.

Contribution

It introduces a novel geometric framework based on Boolean hypercube functions for analyzing measurement incompatibility, offering necessary conditions and tight criteria under certain conditions.

Findings

Complete geometric characterization for unbiased binary qubit measurements

Necessary conditions for biased measurement incompatibility

Falsification of an existing conjecture on measurement incompatibility

Abstract

Deciding which sets of quantum measurements allow a simultaneous readout is a central problem in quantum measurement theory. The problem is relevant not only from the foundational perspective but also has direct applications in quantum correlation problems fueled by incompatible measurements. Although central, only a few analytical criteria exist for deciding the incompatibility of general sets of measurements. This work approaches the problem through functions defined on the Boolean hypercube and their Fourier transformations. We show that this reformulation of the problem leads to a complete geometric characterisation of joint measurability of any finite set of unbiased binary qubit measurements and gives a necessary condition for the biased case. We discuss our results in the realm of quantum steering, where they translate into a family of steering inequalities. When certain unbiasedness conditions are fulfilled, these criteria are tight, hence fully characterizing the steering problem when the trusted party holds a qubit, and the untrusted party performs any finite number of binary measurements. We further discuss how our results point towards a second-order cone programming approach to measurement incompatibility and compare this to the predominantly used semi-definite programming-based techniques. We use our approach to falsify an existing conjecture on measurement incompatibility of special sets of measurements.