Linear Algebra of Generalized Contextuality in All Prepare-Transform-Measure Scenarios

Generalizes the linear-algebraic framework of arXiv:2512.10000, which targeted prepare-measure scenarios, to operational scenarios with sequential transformations of arbitrary stage number, thereby covering all prepare-transform-measure situations. Prior methods for certification and characterization of generalized contextuality were well developed mainly for prepare-measure and single-stage prepare-transform-measure scenarios; this work advances the scope to sequential transformations with an arbitrary number of stages. This is a theoretical extension of an existing framework, explicitly described in the abstract as an extension, and delivered in the form of a full decision procedure.

Provides a full decision procedure for contextuality of such scenarios within operational theories and analyzes its computational complexity. Beyond supplying a decision method, it characterizes complexity: linearly exponential in the minimum generalized probabilistic theory (GPT) dimension and polynomial in the number of procedures. The complexity statement is given in the abstract as an explicit scaling form and constitutes a theoretical analysis result.

Demonstrates the framework through multiple examples, including Spekkens' toy theory and the 8-state single-qubit stabilizer theory, and constructs an operational theory in which contextuality manifests itself only in the sequential structure of the transformations. This construction shows that contextuality can be carried solely by the compositional structure of transformations rather than only by preparation and measurement stages. The examples and construction are theoretical demonstrations; the abstract reports no numerical values or experimental data.

The results target researchers studying generalized contextuality within the operational-theory framework and apply to prepare-transform-measure scenarios with sequential transformations of an arbitrary number of stages; because the decision procedure is linearly exponential in the minimum GPT dimension, the reachable scale is constrained by dimension, while it is polynomial in the number of procedures. The constructed operational theory, in which contextuality manifests only in the sequential structure of transformations, provides an illustrative setting for examining contextuality at the level of compositional structure.

The abstract does not provide the algorithmic details of the decision procedure, the derivation of the complexity bounds, or specific numerical results for the examples; moreover, how general the operational theory with contextuality appearing only in sequential structure is across broader GPTs, and how the framework maps onto experimentally realizable settings, remain open to further examination. Since the available text here is the abstract and bibliographic information without the body, figures, or full derivations, these details cannot be confirmed from this material.