Converting Contextuality into Nonlocality

We introduce a general method which converts, in a unified way, any form of quantum contextuality, including any form of state-dependent contextuality, into a quantum violation of a bipartite Bell inequality. As an example, we apply the method to a quantum violation of the Klyachko-Can-Binicioğlu-Shumovsky inequality.

Figure 1

Every node represents a rank-one projector. A continuous vertical line between $d\ge 3$ nodes indicates that they are mutually orthogonal. Hence, in dimension $d$, in any KS assignment, one of them has to be assigned 1. A dashed line between two nodes indicates that there is a TIFS between (and including) them. Hence, in any KS assignment, both of them cannot be assigned 1. Construction to obtain a critical KS set in dimension $d\ge 3$ from $N\ge d+1$ disjoint bases: (a) For $d=3$ and $N=d+1$. (b) For $d=3$ and $N=d+2$. (c) For $d=3$ and $N=d+3$. (d) For $d=4$ and $N=d+1$. (e) For $d=4$ and $N=d+2$. The construction works similarly for any $d\ge 3$ and $N\ge d+1$. In all cases, it is impossible to assign to the depicted nodes the values 0 or 1 satisfying that one of the $d$ nodes in each continuous vertical line must be 1, while nodes connected by a dashed line cannot both be 1. However, such an assignment is possible whenever we remove any of the depicted nodes.

Figure 2

(a) Five-vertex graph $G$ that represents the relations of orthogonality between the projectors $S=\{{\mathrm{\Pi }}_1,\dots ,{\mathrm{\Pi }}_5\}$ needed to violate the KCBS inequality (5). Projector ${\mathrm{\Pi }}_i$ is represented by vertex $i$, mutually orthogonal projectors are represented by adjacent vertices. (b) Extended 13-vertex graph $\mathcal{G}$ representing the relations of orthogonality between elements of the smallest SI-C set $S^{\prime \prime }=\{{\mathrm{\Pi }}_1,\dots ,{\mathrm{\Pi }}_{13}\}$ that contains $S$. (c) Vertex-weighted graph $(\mathcal{G},w)$ with the weights that produce the largest SI-C. Vertices in white have weight 2 and vertices in black have weight 3. These are the weights used in the SI-C inequality (1) and the Bell inequality (3).