Explaining quantum correlations through evolution of causal models

We propose a framework for the systematic and quantitative generalization of Bell's theorem using causal networks. We first consider the multiobjective optimization problem of matching observed data while minimizing the causal effect of nonlocal variables and prove an inequality for the optimal region that both strengthens and generalizes Bell's theorem. To solve the optimization problem (rather than simply bound it), we develop a genetic algorithm treating as individuals causal networks. By applying our algorithm to a photonic Bell experiment, we demonstrate the trade-off between the quantitative relaxation of one or more local causality assumptions and the ability of data to match quantum correlations.

Figure 1

Sketch of the concept of Pareto optimality for demarcating the boundary between local causality and quantum correlations. In this picture, Bell's theorem rules out the origin only. Our results rule out an entire region of possible models in the presence of relaxations of Bell's assumptions. We rule out this region both rigorously with Theorem t1 and numerically with the evolutionary algorithm that we developed specifically for this task.

Figure 2

Causal networks for Bell-type experiments. On the left is the local hidden variable model, which respects the assumptions going into Bell's famous no-go theorem. Such a model cannot account for certain correlations obtained from measuring entangled particles. The graph in the middle contains a causal link between the measurement settings. Such a model exploits the detection loophole and violates measurement independence. Finally, on the right is a superluminal model which contains a causal link between the outcomes of the experiments.

Figure 3

Photon pairs at 808 nm are emitted via spontaneous parametric down-conversion from a 404 nm pumped bismuth triborate (BiB0) crystal and we prepare polarization enabled photons, labeled “Alice's qubit” and “Bob's qubit.” The components used are half wave plates (HWP), quarter wave plates (QWP), polarizing beam splitters (PBS), polarization maintaining fibers (PMF), and an optical delay line (ODL). Once the state is prepared a polarization tomography setup enables projection of each qubit onto any pure polarization state, which is sufficient to perform two-qubit tomography.

Figure 4

(a) Real component of experimentally measured density matrices for a range $\gamma$ values measured via two-qubit state tomography. (b) The results of the MOEA (run as an EA) on a local model see Fig. 2 to generate the best achievable TVD for various values of $\gamma$ from a Bell-like experiment described in Sec. 5. As theoretically predicted above a certain threshold ($\gamma =\sqrt{2}-1$) the local model can no longer explain the measurement results with zero TVD. This threshold corresponds to violating the CHSH inequality in (15). The linear increase in TVD corresponds to the linear increase in $S$ as discussed in Sec. 5.

Figure 5

Pareto front for the causal network in Fig. 2 using the data from a photonic Bell experiment. The vertical axis labels the causal influence (9) while the horizontal axis labels the closeness to experimental data (7). The blue circles are the values for the nondominated models found by the evolutionary algorithm. For comparison purposes, the straight line is a linear fit to these data.

Figure 6

Pareto front for a local causal network with added $a\to b$ and $x\to \lambda$ edges using the data from a photonic Bell experiment. The vertical axis labels the closeness to experimental data (7). The two horizontal axes label the causal influences (9) for the added edges. The blue circles are the values for the nondominated models found by the evolutionary algorithm. The flat surface is a linear fit to these data.

Figure 7

Example of the causal influence measure $C_{a\to b}\left(I\right)$ given by (9) applied to a more complicated graph. The random variable $y$ is conditioned on and maximized over, as it is a parent of $b$ but not of $a$. By contrast, the variable $x$ is also a parent of $a$ and so it is marginalized over, resulting in the causal influence definition at right.

Figure 8

Island model diagram showing the steps in evolving a three-dimensional Pareto front. This allows the edges of the Pareto front to be found by exploring lower-dimensional graphs with lower populations. Multiple runs in each of step 1 and step 2 can be done concurrently.

Figure 9

3D Pareto front for $a\to b$ and $x\to \lambda$ violations found using the “island model” is shown in blue. Beneath it is a linear mesh which fits the Pareto front with a Pearson's ${\rho }^2$ of 0.9902. By comparison the nonoptimal Pareto front found by combining the best individuals from eight runs using the basic algorithm (i.e., no preevolution) on the full graph is shown in red. The front shown in blue took a fraction of the computing time to find compared to the nonoptimal red runs.

Figure 10

Results of 40 typical runs of the EA with a causal graph allowing “Alice to Bob” local causality violations see Fig. 2. (a) For the purpose of producing this graph, each run had its own $\epsilon$-dominance archive. As can be seen several runs failed to find the correct front at all, indicating that the interplay between hidden nodes and conditioned variables results in a nontrivial search when attempting to match the observed distributions of observable variables. (b) At the conclusion of these 40 runs the $\epsilon$-dominances archive are combined to form the global $\epsilon$-dominance archive generating the best estimate of the actual Pareto. The front is well fit by a straight line (Pearson's ${\rho }^2$ value of 0.997, with bisquare robust fitting). Exact fitting of the distribution (very low TVD) requires additional causality violation. Experimental noise might be the reason for this. Although around 100 runs were conducted to produce the reported results (Fig. 5), very few additional points were found on the Pareto front.