Incompatibility probability of random quantum measurements

TL;DR

This paper investigates the conditions under which quantum measurements are compatible, calculates the probability of incompatibility for random qubit measurements, and provides exact and numerical results relevant to quantum information theory.

Contribution

It derives the exact incompatibility probability for pairs of random qubit measurements and establishes compatibility criteria for collections of quantum measurements.

Findings

Incompatibility probability for two unbiased random qubit measurements is exactly 3/5.

Provides necessary and sufficient conditions for quantum measurement compatibility.

Includes detailed numerical results and figures for general qubit measurement pairs.

Abstract

Incompatibility of quantum measurements is of fundamental importance in quantum mechanics. It is closely related to many nonclassical phenomena such as Bell nonlocality, quantum uncertainty relations, and quantum steering. We study the necessary and sufficient conditions of quantum compatibility for a given collection of $n$ measurements in $d$-dimensional space. From the compatibility criterion for two-qubit measurements, we compute the incompatibility probability of a pair of independent random measurements. For a pair of unbiased random qubit measurements, we derive that the incompatibility probability is exactly $\frac35$. Detailed results are also presented in figures for pairs of general qubit measurements.