Proposed direct test of a certain type of noncontextuality in quantum mechanics

Quantum value indefiniteness 1 refers to the impossibility of a consistent coexistence of certain complementary operationally incompatible quantum observables. It is inferred from three sources: i from quantum violations of constraints on classical probability distributions termed “conditions of possible experience” by Boole 2 , also known as the Boole-Bell-type inequalities 3 , ii from the Kochen-Specker theorem 4–6 , as well as iii from the Greenberger-Horne-Zeilinger 7,8 theorem. Formally, these results are related to the “scarcity” or even total absence of two-valued states identifiable as classical truth assignments on the entire range of quantum observables. In what follows, quantum contextuality 9–13 will be identified with the assertion that the result of a measurement depends on what other observables are comeasured alongside of it. It is one conceivable but not necessary 14 quasiclassical interpretation of quantum value indefiniteness, thereby counterfactually maintaining the “physical existence” of the full domain of possible physical observables. There exist other notions of contextuality based upon violations of some bounds on or conditions imposed by classical probabilities. In their extreme form, these amount to all-ornothing–type contradictions between noncontextual hidden variables and quantum mechanics. The corresponding experimental tests indicate the occurrence of this type of quantum contextuality 15–23 . These findings utilize subsequent measurements of quantum observables contributing to a contradiction with their classical counterparts, but they have no direct bearing on the experiments proposed here which aim at testing another, more direct form of quantum contextuality. A quantum mechanical context 13 is a “maximal collection of comeasurable observables” within the nondistributive structure of quantum propositions. It can be formalized by a single maximal self-adjoint operator, such that every collection of mutually compatible comeasurable operators such as projections corresponding to yes–no propositions are functions thereof 24 , Sec. 84 . Different contexts can be interlinked at one or more common observable s whose Hilbert-space representation is identical and independent of the contexts they belong to. The context independence of the representation of observables by operators e.g., projectors in Hilbert space suggests that quantum contextuality, if it exists, manifests itself in random and uncontrollable single-particle outcomes. A necessary condition for the interlinking of two or more contexts by link observable s is the requirement that the dimensionality of the Hilbert space must exceed two since for lower dimensional Hilbert spaces the maximal operators “decay” into separate isolated “trivial” Boolean sublogics without any common observable. This is also the reason for similar dimensional conditions on the theorems by Gleason, as well as by Kochen and Specker. In what follows we propose an experiment capable of directly testing the contextuality hypothesis via counterfactual elements of physical reality. Indeed, counterfactual reasoning might be considered less desirable than direct measurements as it involves an additional logical inference step rather than a straight empirical finding. In the proposed experiment, two different contexts or, equivalently, two noncommuting maximal observables are simultaneously measured on a pair of spin-one particles in a singlet state 11,25,26 . The contexts are fine tuned to allow a common single observable interlinking them. Although the proposal possesses some conceptual similarities to EinsteinPodolsky-Rosen type experiments, the quantum states as well as the structure of the observables are different. We shall first consider the contexts originally proposed by Kochen and Specker 4 , pp. 71–73 , referring to the change in the energy of the lowest orbital state of orthohelium resulting from the application of a small electric field with rhombic symmetry. The terms Kochen-Specker contexts and maximal Kochen-Specker operators will be used synonymously. More explicitly, the maximal Kochen-Specker operators associated with this link configuration can be constructed from the spin-one observables e.g., Refs. 27,28 in arbitrary directions measured in spherical coordinates