From the Kochen-Specker Theorem to Noncontextuality Inequalities without Assuming Determinism

The Kochen-Specker theorem demonstrates that it is not possible to reproduce the predictions of quantum theory in terms of a hidden variable model where the hidden variables assign a value to every projector deterministically and noncontextually. A noncontextual value assignment to a projector is one that does not depend on which other projectors—the context—are measured together with it. Using a generalization of the notion of noncontextuality that applies to both measurements and preparations, we propose a scheme for deriving inequalities that test whether a given set of experimental statistics is consistent with a noncontextual model. Unlike previous inequalities inspired by the Kochen-Specker theorem, we do not assume that the value assignments are deterministic and therefore in the face of a violation of our inequality, the possibility of salvaging noncontextuality by abandoning determinism is no longer an option. Our approach is operational in the sense that it does not presume quantum theory: a violation of our inequality implies the impossibility of a noncontextual model for any operational theory that can account for the experimental observations, including any successor to quantum theory.

Figure 1

Each of the 18 rays is depicted by a node, and the 9 orthonormal bases are depicted by 9 edges, each enclosing 4 nodes. There is no KS-noncontextual assignment to these nodes. For instance, a noncontextual assignment of $0s$ and $1s$ to 17 of the nodes cannot be completed to an assignment to all 18 because neither 0 nor 1 can be assigned to the remaining node (marked X): one edge enclosing it requires it to be 0, the other 1.

Figure 2

(a) Nine four-outcome measurements, each depicted by a set of four nodes encircled by a blue loop. A yellow hashed region enclosing a set of nodes implies that the corresponding events are operationally equivalent. (b) An illustration of the fact that there is no outcome-deterministic noncontextual assignment to the measurement events. The depicted outcome-deterministic assignment breaks the assumption of noncontextuality for the highlighted pair.

Figure 3

Thirty-six preparation procedures, 9 ensembles of 4 each. A node connected to the elements of an ensemble represents the effective preparation procedure achieved by sampling uniformly from the ensemble.

Figure 4

A noncontextual outcome-indeterministic assignment.