Almost-Quantum Correlations Violate the No-Restriction Hypothesis

To identify which principles characterize quantum correlations, it is essential to understand in which sense this set of correlations differs from that of almost-quantum correlations. We solve this problem by invoking the so-called no-restriction hypothesis, an explicit and natural axiom in many reconstructions of quantum theory stating that the set of possible measurements is the dual of the set of states. We prove that, contrary to quantum correlations, no generalized probabilistic theory satisfying the no-restriction hypothesis is able to reproduce the set of almost-quantum correlations. Therefore, any theory whose correlations are exactly, or very close to, the almost-quantum correlations necessarily requires a rule limiting the possible measurements. Our results suggest that the no-restriction hypothesis may play a fundamental role in singling out the set of quantum correlations among other nonsignaling ones.

Figure 1

Composition of NBFs. (a) A family of bipartite NBFs, such that $\{U_{\alpha |\xi }{\}}_{\alpha }$ defines a Bell measurement for each $\xi$. This collection of Bell measurements defines from the original bipartite box $p(ab|xy)$ a new device with input $\xi$ and output $\alpha$, where $p(\alpha |\xi )=\sum _{a,b,x,y}{U}_{\alpha |\xi }(a,b,x,y)p(a,b,x,y)$. (b) A bipartite NBF $V$ acting on a box $p(\alpha ,c|\xi ,z)$. (c) An implementation of the device in (b) where now the left-hand-side box is realized by the device $U$ of (a). Testing the NBF $V$ in this realization of the box $p(\alpha ,c|\xi ,z)$ effectively tests a tripartite inequality given by the coefficients $W(a,b,c,x,y,z)=\sum _{\alpha ,\xi }V(\alpha ,c,\xi ,z)U_{\alpha |\xi }(a,b,x,y)$. Should the GPT satisfy the no-restriction hypothesis and $\{U_{\alpha |\xi }{\}}_{\alpha }$ and $V$ be AQ NBFs, $W$ would be an AQ NBF.