Measurement-dependent locality beyond independent and identically distributed runs

When conducting a Bell test, it is normal to assume that the preparation of the quantum state is independent of the measurements performed on it. Remarkably, the violation of local realism by entangled quantum systems can be certified even if this assumption is partially relaxed. Here, we allow such measurement dependence to correlate multiple runs of the experiment, going beyond previous studies that considered independent and identically distributed (i.i.d.) runs. To do so, we study the polytope defined by block-i.i.d. measurement-dependent local models. We prove that non-i.i.d. models are strictly more powerful than i.i.d. ones, and comment on the relevance of this work for the study of randomness amplification in simple Bell scenarios with suitably optimized inequalities.

Figure 1

Measurement-dependent local models of a Bell test with $N$ runs, for (a) the i.i.d. case, where all the runs are independent and the same probabilities $P\left(a\right|x\lambda )$, $P\left(b\right|y\lambda )$, $P\left(xy\right|\lambda )$ are used in each run, and (b) the block-i.i.d. case, where an entire stretch of $N$ runs is modelled in parallel as a single block. Notice that the variable $\lambda$ determines not only the local output probabilities $P\left(a\right|x\lambda )$ and $P\left(b\right|y\lambda )$, but also the input probabilities $P\left(xy\right|\lambda )$.

Figure 2

Maximum value of $〈M(1-3h,h)〉$ for ${\mathrm{MDL}}_2$ models in the (a) dependent-runs and (b) independent-runs cases, as shown by the dashed curves. The solid curves represent the value achieved by the quantum point $q_{\text{Hardy}}$. It can be seen that in both cases the ${\mathrm{MDL}}_2$ value exceeds the quantum value at a low threshold value of $h$, indicating that the violation of the inequality in Eq. (E1) by this quantum state is not sufficient to rule out ${\mathrm{MDL}}_2$ models with values of $h$ above this threshold.

Figure 3

Value of $〈M(1-3h,h)〉$ for various independent-runs ${\mathrm{MDL}}_N$ models. The solid curve shows the maximum value that can be achieved by ${\mathrm{MDL}}_2$ models under the uniform-measurements constraint, which, as expected, is lower than the maximum value attainable without the constraint [shown by the dashed curve in Fig. 2, replicated here for comparison]. The dotted curve shows a locus of values of $〈M(1-3h,h)〉$ that we have shown to be attainable by ${\mathrm{MDL}}_N$ models without the uniform-measurements constraint when $N$ is large. This is hence a lower bound on the maximum value achievable by such ${\mathrm{MDL}}_N$ models, and as expected it is already higher than the maximum value for the ${\mathrm{MDL}}_2$ model.