Impossibility of memory in hidden-signaling models for quantum correlations

We consider a toy model for nonlocal quantum correlations in which nature resorts to some form of hidden signaling (i.e., signaling between boxes but not available to the users) to generate correlations. We show that if such a model also had memory, the parties would be able to exploit the hidden signaling and use it to send a message, achieving faster-than-light communication. Given that memory is a resource easily available for any physical system, our results add evidence against hidden signaling as the mechanism behind nature's nonlocal behavior.

Figure 1

Directed acyclic graph depicting the causal structure of the deterministic model. Alice and Bob run a Bell-like experiment by implementing measurements on their distant labs. On each round, a hidden signal that depends on Bob's input and/or output is available to Alice's box. Both boxes also share the value of a hidden variable ${\lambda }_n$.

Figure 2

Two physically equivalent functions $f:{\{0,1\}}^2\to {\{0,1\}}^2$. Left and right relations are different, but they both correspond to the case in which the information being transmitted from Bob's side is $f(y,b)=y\oplus b$. The function is determined once its domain is partitioned, therefore the corresponding values on the target set have no physical meaning.

Figure 3

Directed acyclic graph depicting the causal structure of the hidden signaling scenario when memory effects are present. The variables that enter Alice's device in the round $n-1$ have a causal effect on Alice's output in the $n\mathrm{th}$ round (memory effects are represented by dashed lines). Given that we have traced out hidden variables, edges between nodes should be thought of as probabilistic (rather than deterministic) causal links.

Figure 4

Binomial distributions with parameters $p=\alpha =0.4$ and $N=882$ (green line); and $p=\beta =0.5$ and also $N=882$ (orange line). This value of $N$ is obtained from Eq. (25) for $k=3$. If, out of her $N=882$ odd round outcomes she observes a number of zeros below the intersection point at the center, she has a $99.7\%$ confidence level that Bob is not entering $y=1$ as his input on even rounds. Analogously, she can discard $y=0$ as Bob's input with the same confidence level when the number of zeros observed is above that limit.