“Haunted” quantum contextuality

Two entangled particles in threedimensional Hilbert space (per particle) are considered in an EPR-type arrangement. On each side the Kochen-Specker observables {J 1 , J 2 2 , J 2 3 } and {J̄ 2 1 , J̄ 2 2 , J 2 3 } with [J 2 1 , J̄ 2 1 ] 6= 0 are measured. The outcomes of measurements of J 3 (via J 2 1 , J 2 2 ) and J 2 3 (via J̄ 1 , J̄ 2 2 ) are compared. We investigate the possibility that, although formally J 3 is associated with the same projection operator, a strong form of quantum contextuality states that an outcome depends on the complete disposition of the measurement apparatus, in particular whether J 1 or J̄ 2 1 is measured alongside. It is argued that in this case it is impossible to measure contextuality directly, a necessary condition being a non-operational counterfactuality of the argument. Besides complementarity, contextuality [1, 2, 3, 4, 5] is another, more subtle nonclassical feature of quantum mechanics. That is, one and the same physical observable may appear different, depending on the context of measurement; i.e., depending on the particular way it was inferred. Stated differently, the outcome of a physical measurement may depend also on other physical measurements which are coperformed. In Bell’s own words [1, section 5], “The result of an observation may reasonably depend not only on the state of the system . . . but also on the complete disposition of the apparatus.” This property is usually referred to as contextuality. Formally, contextuality may be related to the nonexistence of two-valued measures on Hilbert logics [6, 7, 8, 9, 10, 11] and the partial algebra of projection operators [12, 13] when the dimension of the Hilbert space is higher than two. Contextuality then expresses the impossibility to construct consistently truth values of the whole physical system by any arrangement of truth values of “proper parts” thereof. The term “proper part” refers to any maximal number of independent comeasurable observables corresponding to commuting self-adjoint operators. In quantum logics [14, 15], these are denoted by boolean