Contextuality Scenario Index

A contextuality scenario specifies the measurements, compatible contexts, and constraints on a noncontextual explanation. This index helps identify which version of contextuality a claim uses. Definitions and general arguments belong to Contextuality.

Sharp-observable scenarios

Kochen–Specker sets. A finite set of rays and orthogonal contexts in dimension at least three is chosen so that no global zero–one assignment satisfies the required basis-sum rules. The obstruction is combinatorial and independent of the prepared state for an appropriate state-independent proof. See Kochen–Specker Theorem.

Peres–Mermin square. Nine observables arranged in compatible rows and columns yield product constraints inconsistent with context-independent values. The contradiction exploits a parity mismatch, not a measurement of all nine observables at once. See Peres–Mermin Square.

KCBS pentagon. Five cyclically exclusive events can be represented by suitable qutrit projectors. A deterministic noncontextual assignment cannot select adjacent vertices, so at most two events receive value one. Therefore

$$\sum_{i=1}^{5}p_i\leq2.$$

The ideal quantum maximum is $\sqrt5$. Experimental use requires the exclusivity and compatibility assumptions of the chosen realization; a generic set of five probabilities does not inherit the bound.

The full KS obstruction and a state-dependent inequality are related but different kinds of result.

Compatibility and unsharp scenarios

Specker-type triangle. Three measurements may be pairwise jointly measurable without admitting one joint measurement for all three. This can occur for generalized measurements. It cannot be realized by three sharp projective measurements that commute pairwise, since those have a common joint spectral measurement.

The compatibility pattern alone is not a complete contextuality test. One must specify the outcome probabilities and the noncontextual model being excluded. Effect Algebras explains why unsharp events require care.

Operational contextuality. Preparations, measurement events, or transformations with identical operational statistics are required to have identical representations in a noncontextual model. The relevant equivalences must be explicitly established, and determinism is not automatically assumed for every unsharp event.

Computational and composite scenarios

Stabilizer and magic-state scenarios. Contextuality is assessed relative to a specified computational subtheory, dimension, and measurement set. State-independent qubit contextuality complicates universal claims that all contextuality equals useful magic. See Contextuality as a Resource.

Bell scenarios. Spatially separated measurements supply compatibility and causal structure. Bell locality can be studied within broader contextual frameworks, but setting independence and spacetime assumptions remain additional physical content. See Nonlocality.

Which entry fits a claim?

Identify whether the argument concerns predetermined sharp values, probabilistic operational representations, compatibility without a global joint measurement, or task advantage. Then identify the ideal equalities that the laboratory procedures must approximate. This prevents a valid result in one scenario from being generalized to another without a proof.

References

• Budroni, Costantino, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson, “Kochen–Specker contextuality,” Reviews of Modern Physics 94, 045007 (2022). Review.
• Klyachko, Alexander A., M. Ali Can, Sinem Binicioğlu, and Alexander S. Shumovsky, “Simple test for hidden variables in spin-1 systems,” Physical Review Letters 101, 020403 (2008). Paper.
• Spekkens, Robert W., “Contextuality for preparations, transformations, and unsharp measurements,” Physical Review A 71, 052108 (2005). Paper.