Quantifying measurement incompatibility of mutually unbiased bases

TL;DR

This paper precisely quantifies the incompatibility of mutually unbiased bases in quantum measurements using noise robustness, providing bounds, demonstrating inequivalence, and exploring applications in quantum steering.

Contribution

It introduces a rigorous quantification of measurement incompatibility for MUBs, including bounds, inequivalence, and implications for quantum steering.

Findings

Tight bounds for complete sets of MUB in prime power dimensions

Existence of operationally inequivalent MUB sets with different noise robustness

Application insights for Einstein-Podolsky-Rosen steering

Abstract

Quantum measurements based on mutually unbiased bases are commonly used in quantum information processing, as they are generally viewed as being maximally incompatible and complementary. Here we quantify precisely the degree of incompatibility of mutually unbiased bases (MUB) using the notion of noise robustness. Specifically, for sets of $k$ MUB in dimension $d$, we provide upper and lower bounds on this quantity. Notably, we get a tight bound in several cases, in particular for complete sets of $k=d+1$ MUB (given $d$ is a prime power). On the way, we also derive a general upper bound on the noise robustness for an arbitrary set of quantum measurements. Moreover, we prove the existence of sets of $k$ MUB that are operationally inequivalent, as they feature different noise robustness, and we provide a lower bound on the number of such inequivalent sets up to dimension 32. Finally, we discuss applications of our results for Einstein-Podolsky-Rosen steering.