[Paper Review] Contextuality without incompatibility

John H. Selby, David Schmid+4 more|arXiv (Cornell University)|Jun 16, 2021

Quantum Mechanics and Applications4 citations

TL;DR

This paper demonstrates that measurement incompatibility is neither necessary nor sufficient for proving the failure of generalized noncontextuality in quantum theory. By showing that any contextuality proof involving incompatible measurements can be recast as a proof using only compatible ones, the authors establish that contextuality arises fundamentally from operational equivalences in unsharp measurements, not incompatibility, thereby redefining the core resource of quantum nonclassicality.

ABSTRACT

The existence of incompatible measurements is often believed to be a feature of quantum theory which signals its inconsistency with any classical worldview. To prove the failure of classicality in the sense of Kochen-Specker noncontextuality, one does indeed require sets of incompatible measurements. However, a more broadly applicable notion of classicality is the existence of a generalized-noncontextual ontological model. In particular, this notion can imply constraints on the representation of outcomes even within a single nonprojective measurement. We leverage this fact to demonstrate that measurement incompatibility is neither necessary nor sufficient for proofs of the failure of generalized noncontextuality. Furthermore, we show that every proof of the failure of generalized noncontextuality in a quantum prepare-measure scenario can be converted into a proof of the failure of generalized noncontextuality in a corresponding scenario with no incompatible measurements.

Motivation & Objective

• To challenge the conventional view that measurement incompatibility is essential for demonstrating quantum contextuality.
• To show that generalized noncontextuality can be violated even in scenarios with only compatible measurements.
• To establish that contextuality arises from operational equivalences in unsharp measurements, not from incompatibility.
• To provide a constructive method for transforming any contextuality proof with incompatible measurements into one with only compatible ones.
• To clarify the foundational role of operational equivalence in nonclassicality, independent of measurement compatibility.

Proposed method

• The authors analyze prepare-measure scenarios where operational equivalences between outcomes of a single POVM (e.g., {½𝕀, ½𝕀}) imply constraints on ontological models.
• They apply the principle of generalized noncontextuality, which demands that operationally equivalent procedures have identical ontological representations.
• They construct a hypercone of noncontextually realizable joint distributions P(y,s) and identify its facet-defining inequalities using linear programming.
• They derive a noncontextuality inequality C ≥ 0, where C = q(p₁|₀ + p₁|₂) + (q−1)p₂|₀ + p₀|₂ − (q+1)p₁|₁, and show quantum realizations violate it.
• They demonstrate that quantum states and POVMs (e.g., M(q) with q ≈ 0.6) yield a violation of C ≥ 0, specifically C ≈ −0.138, ruling out noncontextual models.
• They prove that any contextuality proof involving incompatible measurements can be transformed into an equivalent proof using only compatible measurements, via flag-convexification of the preparation setting.

Experimental results

Research questions

• RQ1 Is measurement incompatibility necessary for proving the failure of generalized noncontextuality in quantum theory?
• RQ2 Can contextuality be demonstrated in a prepare-measure scenario without any incompatible measurements?
• RQ3 What role do operational equivalences between outcomes of a single unsharp measurement play in generating contextuality?
• RQ4 Can every proof of contextuality involving incompatible measurements be recast as a proof using only compatible measurements?
• RQ5 What is the fundamental resource underlying contextuality—measurement incompatibility or operational equivalence in unsharp measurements?

Key findings

• The paper proves that measurement incompatibility is not necessary for contextuality: any contextuality proof involving incompatible measurements can be transformed into one using only compatible measurements.
• A specific violation of the noncontextuality inequality C ≥ 0 is demonstrated, with C ≈ −0.138 for the quantum realization M(q), which rules out generalized-noncontextual models.
• The existence of nontrivial operational equivalences between outcomes of a single unsharp POVM (e.g., {½𝕀, ½𝕀}) leads to nontrivial constraints on ontological models, enabling contextuality without incompatibility.
• The result shows that contextuality is not inherently tied to incompatibility but stems from the structure of operational equivalences in unsharp measurements.
• The framework of generalized noncontextuality can be applied to single unsharp measurements, revealing nontrivial constraints that lead to contextuality even in the absence of measurement incompatibility.
• The authors refute the claim in Ref. [25] that incompatibility is both necessary and sufficient for contextuality, showing it fails under realistic assumptions like restricted state sets.