Quantum nonlocality in the presence of strong measurement dependence

It is well known that the effect of quantum nonlocality, as witnessed by violation of a Bell inequality, can be observed even when relaxing the assumption of measurement independence, i.e., allowing for the source to be partially correlated with the choices of measurement settings; however, what is the minimal amount of measurement independence needed to observe quantum nonlocality? Here we explore this question and consider models with strong measurement-dependent locality, where measurement choices can be perfectly determined in almost all rounds of the Bell test. Nevertheless, we show that quantum nonlocality can still be observed in this scenario, which we conjecture is minimal within the framework we use. We also discuss potential applications in randomness amplification.

Figure 1

We consider a Bell test with relaxed measurement independence. Specifically, the source, i.e., the classical variable $\mathrm{\Lambda }$, can be correlated to the choice of measurement inputs (variables $X$ and $Y$) of both parties. We show that quantum nonlocality can exhibit strong measurement-dependent nonlocality, i.e., the quantum predictions cannot be explained by a local model even if the source $\mathrm{\Lambda }$ completely determines both inputs in almost all rounds of the experiment.

Figure 2

Graph depicting nonlocality in the presence of a strong measurement dependence attained by the states $|{\varphi }_v\rangle \langle {\varphi }_v|=v|{\varphi }_{+}\rangle \langle {\varphi }_{+}|+(1-v)\mathbb{1}/16$, where $|{\varphi }_{+}\rangle ={\sum }_{j=0}^3|jj\rangle /2$. The measurements correspond to those used to obtain the maximal violation of the Mermin-Peres magic-square game. For given visibility $v$, nonlocality is observed if the values of the parameters $\xi$ and $q$ belong to the region above the corresponding curve. The parameter $\eta$ is taken to be equal to $2/9-\xi$. The red bold curve corresponds to a visibility comparable to that in the experiment of Ref. [27], based on a hyperentangled state.