Quantum Nonlocality and Device-Independent Randomness Robust to Relaxations of Bell Assumptions

TL;DR

This paper demonstrates that quantum nonlocality and device-independent randomness can be certified under relaxed Bell assumptions using states of constant local dimension, with implications for practical quantum information protocols.

Contribution

It shows that constant-dimensional states suffice for nonlocality certification under relaxed assumptions and introduces a new framework for analyzing behaviors with measurement and parameter dependence.

Findings

Constant local dimension states certify nonlocality under relaxations.

Characterization of behaviors under measurement and parameter dependence.

Analytical quantum guessing probability as a function of noise and leakage.

Abstract

The question of certifying quantum nonlocality under a relaxation of the assumptions in the Bell theorem has gained traction, with potential for device-independent applications under weak seeds and cross-talk. Recently, it was shown that quantum nonlocality can be certified even under a simultaneous arbitrary (but not full) relaxation of the assumptions of Measurement Independence (MI) and Parameter Independence (PI), using states of local dimension $d = poly((1-\epsilon)^{-1})$ for an $\epsilon \in [0,1)$-relaxation. Here, we derive three results strengthening the state-of-art. Firstly, we show that states of constant local dimension $d$ are already sufficient to certify quantum nonlocality under arbitrary MI and PI relaxation, albeit in a non-robust manner. Secondly, and as a theoretical paradigm to derive the above, we introduce the notion of \textit{measurement-dependent parameter-dependent locality} as the set of input-output behaviors under simultaneous relaxations of measurement and parameter independence. We provide a rigorous characterisation of the vertices of the polytope of joint input-output behaviors that obey a $\mu$-relaxation of MI and $\epsilon$-relaxation of PI. We highlight a relation between nonlocality certification under PI relaxation and that under detection inefficiencies by pointing out alternative extremal correlations to the Eberhard correlations that also allow to achieve detection efficiency of $\eta = 2/3$ in the two-input scenario. Finally, we study the implication of the relaxed assumptions for device-independent randomness certification. We analytically derive the quantum guessing probability for one player's outcomes in the CHSH Bell test, as a function of the noise in the test as well as of a leakage of an average amount of $I(X:B) < 1$ bits of input information per measurement round.