Exclusivity principle and unphysicality of the Garg-Mermin correlation

The question concerning the physical realizability of a probability distribution is quite important in quantum foundations. Recent studies emphasized a principle which was first indicated by Specker. This principle, called Specker's principle, bounds physically realizable correlations. We study Specker's observation in the scenario that involves three inputs each with two outputs. Then using only linear constraints imposed on joint probabilities by this principle, we reveal the unphysical nature of Garg-Mermin (GM) correlation. Interestingly, GM correlation was proposed to falsify the following suggestion by Fine: if the inequalities of Clauser and Horne (CH) holds, then there exists a deterministic local hidden-variable model for a spin-1/2 correlation experiment of the Einstein-Podolsky-Rosen type, even when more than two observables are involved on each side. Our result establishes that, unlike in the CH scenario, the local orthogonality principle at single copy level is not equivalent to the no-signaling condition in the GM scenario.

Figure 1

(a) Single-party black box scenario. $x_i$ denotes input and correspondingly $o_i$ the output. The exclusivity principle concerns the scenario whenever a joint distribution $\left\{P(o_1,...,o_k|x_1,...,x_k)\right\}$ exists for a set of inputs ${\left\{x_i\right\}}_{i=1}^k$. In general the output $o_i$ are multivalued, but here we consider $o_i\in \{0,1\}$. (b) Two-party black box scenario. Joint distributions of the type $\left\{P\left(o_i^A{o}_j^B|x_i^A{x}_j^B\right)\right\}$ always exist for all $i,j$.