No-Hypersignaling Principle

A paramount topic in quantum foundations, rooted in the study of the Einstein-Podolsky-Rosen (EPR) paradox and Bell inequalities, is that of characterizing quantum theory in terms of the spacelike correlations it allows. Here, we show that to focus only on spacelike correlations is not enough: we explicitly construct a toy model theory that, while not contradicting classical and quantum theories at the level of spacelike correlations, still displays an anomalous behavior in its timelike correlations. We call this anomaly, quantified in terms of a specific communication game, the “hypersignaling” phenomena. We hence conclude that the “principle of quantumness,” if it exists, cannot be found in spacelike correlations alone: nontrivial constraints need to be imposed also on timelike correlations, in order to exclude hypersignaling theories.

Figure 1

Spacelike and timelike correlations. Events $A$ and $B$ are spacelike separated; i.e., information cannot travel from the one to the other (no-signaling principle). Correspondingly, they can only share spacelike correlations, previously distributed in the form of a bipartite state $\mathrm{\Omega }$. Events $A$ and $C$ are timelike separated; i.e., information can indeed travel from $A$ to $C$: such information is encoded into the states $\{{\mathrm{\Omega }}_x\}$, and later decoded by the measurement $\{E_y\}$. As the no-signaling principle constrains spacelike correlations, the no-hypersignaling principle constrains timelike correlations.

Figure 2

Illustration of a hypersignaling theory. While the system $S$ alone satisfies ${\mathcal{P}}_S^{m\to n}\subseteq {\mathcal{P}}_d^{m\to n}$, and thus has signaling dimension $d$, the composite system $S\otimes S$ has a signaling dimension strictly larger than $d^2$.

Figure 3

No-hypersignaling vs information causality and vs local tomography. Left: the diagram compares theories satisfying information causality (yellow set) and the no-hypersignaling principle (blue set): CT (classical theory), QT (quantum theory), PR model (the toy model theory for PR boxes), and HS model (the locally classical, hypersignaling theory constructed in this Letter). Right: comparison between local tomography and no hypersignaling as two features of general probabilistic theories. Examples of theories that are nonhypersignaling but violate local tomography are provided by real quantum theory (RQT) and fermionic quantum theory (FQT). The HS model is locally tomographic but hypersignaling. Finally CT, QT, and the PR model lie in the intersection, as they obey both local tomography and the no-hypersignaling principle.