[Paper Review] Contextuality beyond the Kochen-Specker theorem

Ravi Kunjwal|arXiv (Cornell University)|Dec 21, 2016

Quantum Mechanics and Applications75 references11 citations

TL;DR

This thesis extends contextuality beyond the Kochen-Specker theorem by developing an operational framework for contextuality that does not rely on deterministic hidden variables. It introduces noncontextuality inequalities without assuming determinism, generalizes contextuality to n-cycle scenarios, and establishes a bridge between Specker’s scenario and broader quantum foundations, enabling experimental tests of contextuality in single-system scenarios without Bell-type assumptions.

ABSTRACT

When it isn't possible to tell two distinct experimental procedures apart purely from their input/output statistics, then it seems a plausible hypothesis that the two procedures must be physically identical. We call such a hypothesis "noncontextuality", an instance of Leibniz's principle of the identity of indiscernibles. Read in the contrapositive, this hypothesis entails that any physical distinctions between two experimental procedures must necessarily lead to a difference in their operational statistics. The results I present in this thesis concern the failure of this hypothesis -- a failure dubbed "contextuality" -- when one tries to embed an operational theory (such as quantum theory) in the ontological models framework. The Kochen-Specker theorem demonstrates the failure of noncontextuality for deterministic ontological models of quantum theory, i.e., those ontological models where the ontic/physical state of the system fixes the outcome of any projective measurement on the system in a deterministic manner. This thesis goes beyond the Kochen-Specker (KS) theorem by asking what operational facts must be verified in experiments to conclude that Nature does not admit noncontextual ontological models, not even indeterministic ones. This leads to noncontextuality inequalities that are robust to noise in the preparations and measurements. In the particular case of quantum theory, these inequalities are meaningful even when unsharp measurements (or POVMs) are allowed, a feature not shared by the traditional approach to KS-noncontextuality where unsharp measurements are excluded by fiat: allowing them renders even trivial POVMs (proportional to identity) maximally KS-contextual. The sense in which trivial POVMs are indeed "trivial" (or "noncontextual") is clear in our approach: they are simply too noisy to lead to violation of our noncontextuality inequalities.

Motivation & Objective

• To develop a framework for contextuality that moves beyond the assumptions of the Kochen-Specker theorem, particularly the assumption of deterministic hidden variables.
• To generalize contextuality to arbitrary n-cycle compatibility scenarios, extending Specker’s original parable of a noncontextual assignment.
• To derive noncontextuality inequalities without assuming determinism, enabling experimental tests of contextuality in single-system scenarios.
• To unify operational and graph-theoretic approaches to contextuality, particularly connecting with Spekkens’ operational framework and the graph-theoretic characterization of contextuality scenarios.
• To explore contextuality as a resource in quantum information, including robustness to noise and its role in measurement-based quantum computation.

Proposed method

• Adapts Spekkens’ operational framework for contextuality to derive noncontextuality inequalities without assuming determinism, using joint measurability structures.
• Generalizes contextuality to n-cycle compatibility scenarios by modeling compatibility relations as graphs and analyzing the resulting noncontextuality inequalities.
• Employs graph-theoretic tools to characterize contextuality scenarios, linking them to the Kochen-Specker graph framework and extending it to nonprojective observables.
• Uses the framework to analyze the robustness of contextuality under noise, defining 'contextuality-breaking' channels analogous to those for entanglement and nonlocality.
• Applies the formalism to derive experimental tests of contextuality in single-system scenarios, such as the one demonstrated in the Mazurek et al. (2015) experiment.
• Establishes connections between contextuality and other quantum features by analyzing epistemic restrictions and their limits in classical simulations.

Experimental results

Research questions

• RQ1 Can contextuality be formulated in an operational framework that does not assume deterministic hidden variables, as in the Kochen-Specker theorem?
• RQ2 How can contextuality be generalized from Specker’s scenario to arbitrary n-cycle compatibility structures?
• RQ3 What are the noncontextuality inequalities that can be derived in the absence of determinism, and how do they enable experimental tests?
• RQ4 What is the relationship between Bell nonlocality and contextuality beyond the folklore that nonlocality is a special case of contextuality?
• RQ5 Can contextuality be quantified as a resource in quantum information, and how robust is it under noise and decoherence?

Key findings

• The thesis derives noncontextuality inequalities without assuming determinism, enabling experimental tests of contextuality in single-system scenarios, as demonstrated in the Mazurek et al. (2015) experiment.
• It generalizes contextuality to n-cycle compatibility scenarios, providing a systematic framework for analyzing contextuality in arbitrary cyclic joint measurability structures.
• The work establishes a quantitative bridge between the graph-theoretic characterization of contextuality and Spekkens’ operational framework, unifying two major approaches.
• It identifies that contextuality can be robustly analyzed under noise, with the potential to define 'contextuality-breaking' channels analogous to those for entanglement and nonlocality.
• The framework enables the detection of contextuality in scenarios involving nonprojective observables, extending the scope of the graph-theoretic approach.
• The results support the view that contextuality is a fundamental resource for quantum advantage, particularly in measurement-based quantum computation, and can be certified without Bell-type assumptions.