Symmetric observations without symmetric causal explanations

Abstract

Inferring causal models from observed correlations is a challenging task, crucial to many areas of science. In order to alleviate the effort, it is important to know whether symmetries in the observations correspond to symmetries in the underlying realization. Via an explicit example, we answer this question in the negative. We use a tripartite probability distribution over binary events that is realized by using three (different) independent sources of classical randomness. We prove that even removing the condition that the sources distribute systems described by classical physics, the requirements that i) the sources distribute the same physical systems, ii) these physical systems respect relativistic causality, and iii) the correlations are the observed ones, are incompatible.

Figures (4)

• Figure 1: \ref{['fig:triangleDAG']} The triangle causal structure. Three bipartite latent variables, $\alpha$, $\beta$ and $\gamma$ influence each a different set of two out of the three visible variables $A$, $B$ and $C$. This structure has received considerable attention in the context of nonlocality in quantum networks. \ref{['fig:setting']} Pictorial representation of the relations between realizations and observations. In this work we demonstrate that $\textbf{?}=\text{✗}$. Namely, there exist probability distributions over the binary-valued observations that i) are produced in the triangle causal structure by classical physical systems and ii) are invariant under permutations of the visible nodes, yet they do not admit explanations in terms of symmetric models, i.e., when $\alpha$, $\beta$ and $\gamma$ denote copies of a same physical system.
• Figure 2: \ref{['fig:triangle']} simplified representation of the triangle DAG in Fig. \ref{['fig:triangleDAG']} for symmetric causal scenarios. Each line represents an independent copy of a same bipartite state $\omega$, that is distributed to the two parties (the nodes) that it is connected to. All parties perform the same measurement, given by the effect $\{e^o\}_o$, to produce their outcomes. Any process following this causal mechanism leads to distributions over the observable events that are invariant under permutations of the observable nodes. For all observations that are produced in this way, one can consider the distributions that are created when using more copies of the state and the measurement device like, for instance, that in \ref{['fig:inflation']}. We use the consistency conditions implied by the existence of this distribution to prove that the distribution with $E_1=E_1^c$, $E_2=-1/3$, and $E_3=E_3^c$ is impossible to generate in the symmetric triangle network under any physical theory consistent with relativity.
• Figure 3: Depiction of the hierarchy of inflations that we use. In each level, we demand the existence of probability distributions for each of the networks with smaller number of nodes too. These distributions are related to each other by constraints on the marginals over a same number of nodes (see Eq. \ref{['eq:iden']}).
• Figure 4: Projection of the space of probability distributions given by Eq. \ref{['eq:prob']} on the plane defined by $E_1$ and $E_2$. The points in the gray area produce negative probabilities for any value of $E_3$. The points in the blue area correspond to distributions that cannot be generated in the triangle scenario, regardless of the nature of the sources gisin2020. In the area to the left of the yellow boundary for every $E_1$ and $E_2$ there exists at least one value of $E_3$ for which the corresponding distribution admits a realization in the triangle structure with classical latent variables gisin2020. In particular, the blue triangle denotes the distribution that we use in the work, with $E_1=E_1^c\approx0.1753$ and $E_2=-1/3$. The orange area denotes the distributions that are proven not to admit a symmetric realization in the triangle scenario using the 11th level of the corresponding inflation hierarchy using the constraints of the type of Eq. \ref{['eq:iden']}. Using $\sim50$ GB of RAM and one day of compute we are able to identify as incompatible also the point denoted with a red circle ($E_1=0.1580$ and $E_2=-1/3$) using the 15th level of the hierarchy. The dark green area contains the distributions that are identified as incompatible by the certificate obtained by substituting the constraints \ref{['eq:iden']} by \ref{['eq:nolpi']} and testing the distribution with $E_1=0.1656$ and $E_2=-1/3$. The expression of the certificate is given in Appendix \ref{['app:cert']}.