Robust Bell inequalities from communication complexity

TL;DR

This paper explores the construction of robust Bell inequalities resistant to noise and detection loopholes, deriving large quantum violations from communication complexity gaps, with implications for quantum nonlocality tests.

Contribution

It introduces a method to derive large violations of Bell inequalities from communication complexity bounds, including inefficiency-resistant inequalities, enhancing robustness against noise and loopholes.

Findings

Large violations can be exponentially derived from communication complexity gaps.

Inefficiency-resistant Bell inequalities can be constructed with violations exponential in input size.

The approach improves robustness of Bell tests against noise and detection loopholes.

Abstract

The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.