Unified approach to contextuality, nonlocality, and temporal correlations

We highlight the existence of a joint probability distribution as the common underpinning assumption behind Bell-type, contextuality, and Leggett-Garg-type tests. We then present a procedure to translate contextual scenarios into temporal Leggett-Garg-type and spatial Bell-type ones. To demonstrate the generality of this approach we construct a family of spatial Bell-type inequalities. We show that in the Leggett-Garg scenario a necessary condition for contextuality in time is given by a violation of consistency conditions in the consistent histories approach to quantum mechanics.

Figure 1

Pictorial description of the histories related to the interference terms in Eq.(24): (a) histories $\left\{\right(+,k,k),(-,k,k\left)\right\}$, related to the term $\langle X_2{X}_3\rangle$, can give rise to a nonzero interference term $I(*,k,k)$; (b) histories $\left\{\right(k,+,k),(k,-,k\left)\right\}$, related to the term $\langle X_1{X}_3\rangle$, can give rise to a nonzero interference term $I(k,*,k)$; and (c) histories $\left\{\right(k,k,+),(k,k,-\left)\right\}$, related to the term $\langle X_1{X}_2\rangle$, are always consistent, therefore there is no interference of the form $I(k,k,*)$.