Bell tests with min-entropy sources

TL;DR

This paper investigates how relaxing measurement independence assumptions in Bell tests affects the certification of quantum non-locality, deriving bounds on min-entropy and exploring the implications for device-independent protocols.

Contribution

It introduces a new framework for analyzing measurement dependence using min-entropy bounds and compares it to Santha-Vazirani sources, providing tighter constraints.

Findings

Lower bounds on min-entropy of measurement settings are established.

Relaxing measurement independence leads to more restrictive bounds than Santha-Vazirani assumptions.

A procedure for finding tight bounds on measurement dependence is outlined.

Abstract

Device independent protocols rely on the violation of Bell inequalities to certify properties of the resources available. The violation of the inequalities are meaningless without a few well-known assumptions. One of these is measurement independence, the property that the source of the states measured in an inequality is uncorrelated from the measurements selected. Since this assumption cannot be confirmed, we consider the consequences of relaxing it and find that the definition chosen is critically important to the observed behavior. Considering a definition that is a bound on the min-entropy of the measurement settings, we find lower bounds on the min-entropy of the seed used to choose the inputs required to deduce any quantum or non-local behavior from a Bell inequality violation. These bounds are significantly more restrictive than the ones obtained by endowing the seed with the further structure of a Santha-Vazirani source. We also outline a procedure for finding tight bounds and study the set of probabilities that can result from relaxing measurement dependence.