Causal Networks and Freedom of Choice in Bell’s Theorem

Bell’s theorem is typically understood as the proof that quantum theory is incompatible with local-hidden-variable models. More generally, we can see the violation of a Bell inequality as witnessing the impossibility of explaining quantum correlations with classical causal models. The violation of a Bell inequality, however, does not exclude classical models where some level of measurement dependence is allowed, that is, the choice made by observers can be correlated with the source generating the systems to be measured. Here, we show that the level of measurement dependence can be quantitatively upper bounded if we arrange the Bell test within a network. Furthermore, we also prove that these results can be adapted in order to derive nonlinear Bell inequalities for a large class of causal networks and to identify quantumly realizable correlations that violate them.

Popular Summary

Bell's theorem is a cornerstone in our understanding of quantum theory. Apart from its radical foundational consequences, it is a resource in a variety of tasks, from cryptography and self-testing to distributed computing. However, because of stringent constraints, such as locality and detection efficiency, testing those ideas experimentally has always been a thorny issue. Reason why is that only fifty years later, Bell seminal results have finally been proved in a loophole-free manner. Nonetheless, a seemingly untestable loophole has yet remained: the “free will” or measurement independence assumption, stating that the choice of which measurement to perform in a physical system is independent of properties of such a system. Can this assumption in Bell's theorem ever be tested? By embedding Bell's original causal structure into a larger causal network, we positively solve this question.

We obtain lower and upper bounds to the level of measurement dependence of a Bell experiment. From that we derive new Bell inequalities–which explicitly incorporate correlations between the system to be measured and the settings of measurement devices–the violation of which is an unambiguous signature of nonclassicality. As a byproduct, we establish a general connection between Bell scenarios with measurement dependence and causal networks of growing size and complexity, proving, for the first time, that they can lead to nonclassical correlations.

Our results are a first step, but the tools and concepts we introduce offer a clear way to tackle the many open questions in the study of quantum networks and the role of causality in quantum theory.

Figure 1

Bell’s causal structure. Two distant parties receive their shares of a joint subsystem prepared by a common source $\mathrm{\Lambda }$. The variables $X$ and $Y$ represent the measurement choices and $A$ and $B$ the corresponding measurement outcomes for the observers Alice and Bob, respectively.

Figure 2

Bell’s causal structure with measurement dependence. The purple arrows represent the fact that the correlations between $X$, $Y$, and $\mathrm{\Lambda }$ can be due to some direct causal influence among them or mediated by an external common source. We emphasize that even though the settings are plausibly common-cause connected with $\mathrm{\Lambda }$, typically one can also be confident that they are also (strongly) influenced by independent localized sources of randomness.

Figure 3

A causal structure assessing the measurement dependence. The extra measurement outcome variable $R$ can be used to provide an upper bound on the measurement dependence. Note that this model allows, at least in principle, for arbitrary correlations between the input variables $X$ and $Y$ and the source $\mathrm{\Lambda }$.

Figure 4

The causal structure of a standard multipartite Bell scenario. Each of the $n$ distant parties has a common source $\mathrm{\Lambda }$ and inputs and outputs labeled as $X_i$ and $A_i$, respectively.

Figure 5

A causal structure assessing measurement dependence in a multipartite Bell scenario. The auxiliary variable $R$ allows one to practically upper bound the amount of measurement dependence that might be present. The double-arrowed (purple) edges indicate that the correlations between the input variables $X_i$ and $\mathrm{\Lambda }$ might arise from direct causal influence or via a common source.

Figure 6

Measurement dependence and Bell-inequality violations. The orange curve shows the lower bound on the measurement dependence $I(X,Y:\mathrm{\Lambda })$ as described by Eq. (12), needed to explain a given degree of quantum violation for the CHSH inequality given by $(\mathrm{CHSH}-2)/2\sqrt{2}$. The blue curve shows the lower bound on the measurement dependence $I(X,Y,Z:\mathrm{\Lambda })$ as described by Eq. (36), needed to explain a given degree of quantum violation for the Mermin inequality given by $(\mathrm{M}-2)/4$. A comparison between the two curves shows that a higher degree of measurement dependence is required to explain the violation of the Mermin inequality with the same degree of quantum violation as the CHSH case, a point that can be of experimental relevance when trying to violate measurement-dependent Bell inequalities.

Figure 7

The triangle network. Pairwise-independent sources generate the correlations between the three observable variables $\alpha$, $\beta$, and $R$.

Figure 8

The “2’s & $\mathit{n}$” network for $n=3$. One observable source connects three of the four observable variables; however, the other sources only connect pairs. More generally, a “2’s & $n$” consists of observable variables ${\alpha }_1,\dots ,{\alpha }_n$ in addition to $R$. Note that the triangle network (see Fig. 7) is a “2’s & $n$” network with $n=2$.

Figure 9

A cyclic causal structure with $n=4$. Each observable variable its connected to its neighbor via a common source. Note that the triangle network (see Fig. 7) is a particular case of a cyclic network with $n=3$. We index the observable variables in an $n$-cyclic scenario as ${\alpha }_1,\dots ,{\alpha }_{n-1}$ along with $R$.

Figure 10

A bilocality causal structure, shown here with measurement dependence (sources and local inputs potentially correlated) along with an auxiliary variable $R$. In the absence of measurement dependence, this scenario is akin to an entanglement-swapping scenario [75], with a central node sharing correlations with two peripheral nodes via two independent sources. More generally, in an $n$-locality scenario with measurement dependence and an auxiliary variable, we have a chain of outcome variables $A_1,\dots ,A_{n+1}$ along with $R$ and only the two peripheral nodes $A_1$ and $A_{n+1}$ have inputs.