From the Kochen-Specker theorem to noncontextuality inequalities without assuming determinism (1506.04150v1)

Abstract: 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.

An Examination of Kochen-Specker Theorem Generalization to Noncontextuality Inequalities

The paper, "From the Kochen-Specker theorem to noncontextuality inequalities without assuming determinism," addresses a significant topic in quantum mechanics: the implications of the Kochen-Specker (KS) theorem concerning hidden variable models and noncontextuality. This inquiry extends traditional interpretations by relaxing the assumption of deterministic value assignments to projectors, subsequently yielding generalized noncontextuality inequalities.

Summary

The KS theorem illustrates the impossibility of replicating quantum predictions through hidden variable models where variables assign deterministic and noncontextual values to projectors. Noncontextuality here posits that the value assigned by hidden variables to a projector should not depend on other co-measured projectors. Extending beyond deterministic assumptions, the authors introduce a scheme that examines whether experimental statistics comply with a noncontextual model, denoting this broader framework as universal noncontextuality.

In KS models, context-independence leads to deterministic assignments. By allowing outcome-indeterministic models, measurement noncontextuality introduces scenarios where probabilities rather than deterministic values are context-independent. This theoretical renovation facilitates robust noncontextuality inequalities, overcoming the inadequacies of earlier models that presumed determinism.

Strong Numerical Results

The authors derive an operationally significant inequality for a quantum setup involving 18 rays in a four-dimensional Hilbert space arranged in nine orthonormal bases. The paper illustrates the infeasibility of noncontextual deterministic assignments for these rays—a characteristic of the KS theorem—via hypergraph representation. The central operational inequality established is:

$$A \equiv \frac{1}{36} \sum_{i=1}^{9} \sum_{k=1}^{4} p(k|M_{i},P_{i,k}) \le \frac{5}{6}.$$

In quantum realization, with perfect predictability, $A$ achieves its maximal value of 1, contradicting the noncontextual bound.

Theoretical and Practical Implications

This work has both theoretical and experimental consequences for noncontextual models in physics. The proposition of noncontextual inequalities paves the way for empirical validation of contextuality assumptions, bringing quantum theory interpretations closer to experimentally observable phenomena. It implies that successor theories to quantum mechanics must also conform to or violate such constraints.

The results significantly impact our understanding of quantum foundations, suggesting that noncontextual models are less capable of capturing quantum reality's underlying nature, especially under ideal experimental conditions with suppressed noise.

Speculation on Future Developments

The methodology exemplified here may guide future studies in designing experiments sensitive to context-dependence, even amid imperfections. This trajectory could explore new models of quantum phenomena, integrate theoretical predictions with experimental realities more deeply, and refine our interpretations of quantum mechanics.

To conclude, the paper invites a reconsideration of noncontextuality in quantum mechanics, applying rigorous comparative analyses and unlocking pathways to innovative empirical tests that confront entrenched theoretical paradigms in quantum foundations. The pragmatic approach and mathematical rigor inherent in these noncontextual inequalities could inform next-generation quantum theories and physics models that transcend traditional contextual assumptions.