Monogamy relations for relativistically causal correlations

TL;DR

The paper investigates how relativistic causality constrains multi-party correlations beyond the standard non-signalling framework. It develops an entropic approach that yields strong monogamy relations among space-like separated influences and provides a general method (via the Shannon cone and Fourier-Motzkin elimination) to derive such inequalities. The results show that relativistically causal correlations are highly non-local and that proposed jamming mechanisms enabling such correlations would inevitably enable superluminal signaling, challenging causal explanations and impacting cryptographic considerations. By extending entropy-based tools to relativistic causality, the work offers a versatile framework for distinguishing between non-signalling and relativisticcausal correlations in complex networks, with potential implications for theory and experimentation alike.

Abstract

Non-signalling conditions encode minimal requirements that any (quantum) systems must satisfy in order to be consistent with special relativity. Recent works have argued that in scenarios involving more that two parties, correlations compatible with relativistic causality do not have to satisfy all possible non-signalling conditions but only a subset of them. Here we show that correlations satisfying only this subset of constraints have to satisfy highly non-local monogamy relations between the effects of space-like separated random variables. These monogamy relations take the form of entropic inequalities between the various systems and we give a general method to derive them. Using these monogamy relations we refute previous suggestions for physical mechanisms that could lead to relativistically causal correlations, demonstrating that such mechanisms would lead to superluminal signalling.

Figures (3)

• Figure 1: Contrasting multi-partite generalisations of non-signalling conditions. Three-party scenario involving Alice, Bob and Charlie with inputs (outputs) $x, (a)$, $y, (b)$ and $z, (c)$ respectively. The parties are arranged on a line and such that their choice of input and generation of output happens space-like separated, as illustrated with the three future light cones. As the intersection of the light cones $\mathcal{K}_{(x,a)}$ and $\mathcal{K}_{(z,c)}$ lies fully within $\mathcal{K}_{(y,b)}$, the relativistic causality constraints do not include $P(ac|xyz)=P(ac|xz)$$\forall \ a,c,x,y,z$, which is among the usual non-signalling constraints Masanes. Notice that this means that Bob's input can only affect the correlations of Alice and Charlie not their marginals, i.e., $P(a|xyz)=P(a|x)$, $P(c|xyz)=P(c|z)$$\ \forall \ a,c,x,y,z$. One way to achieve such an influence is by means of the functional dependency $A\oplus C =Y$, which will be used throughout this work.
• Figure 2: Scenario in multiple spatial dimensions. The three parties in orange can choose inputs $X$, $Y$, $Z$ and obtain outcomes $X'$, $Y'$, $Z'$ while Alice, Bob and Charlie choose inputs $A'$, $B'$, $C'$ and obtain outcomes $A$, $B$ and $C$ respectively. For the correlations considered in the main text, only $A,B,C,X,Y,Z$ play a role. The six events where these variables are generated are space-like separated and are displayed here in a frame where they are simultaneous (see (b)).
• Figure 3: (a) Compass setup. The two parties Xavier and Yanina (orange) choose inputs $(X, X_m)$, $(Y, Y_m)$ while Alice, Bob and Charlie (blue) generate outputs $A$, $B$ and $C$ respectively. The five events are all space-like separated and are displayed here in a frame where they are simultaneous. (b) Line setup. This scenario, compared to the compass, does not feature the constraint $P(ac|xy)=P(ac)$$\forall \ a,c,x,y$, while the others remain the same.

Theorems & Definitions (4)

• Definition 2.1: Relativistic causality constraints
• Example 4.1
• Example 4.2
• Example Supplementary Note 1.1