Bell inequality violations with random mutually unbiased bases

TL;DR

This paper investigates the likelihood of Bell inequality violations in high-dimensional entangled states using random mutually unbiased bases, showing near-guaranteed violations with minimal measurement settings.

Contribution

It demonstrates that Bell violations can be reliably observed with random MUBs in higher dimensions, even with limited measurement settings, highlighting experimental robustness.

Findings

Near-guaranteed Bell violations for maximally entangled qutrits and ququarts.

High probability (~99.84%) of Bell violation in maximally entangled ququints.

All no-violation cases still violate some more-setting Bell inequalities.

Abstract

We examine the problem of exhibiting Bell nonlocality for a two-qudit entangled pure state using a randomly chosen set of mutually unbiased bases (MUBs). Interestingly, even if we employ only two-setting Bell inequalities, we find a significant chance of obtaining a Bell violation if the two parties are individually allowed to measure a sufficient number of MUBs. In particular, for the case of maximally entangled qutrits and ququarts, our numerical estimates indicate that we can obtain near-guaranteed Bell violation by considering only such Bell inequalities. The case of maximally entangled ququints is similar, albeit the chance of ending up with a successful trial decreases somewhat to approximately $99.84\%$. Upon a closer inspection, we find that even all these no-violation instances violate some more-setting Bell inequalities. These results suggest that the experimental tests of Bell nonlocality for these higher-dimensional entangled states remain viable even if the two parties do not share a common reference frame.