Genuine Global Kochen-Specker Contextuality as Classical Coordination Cost

Abstract: Classical simulations of quantum correlations can fail because no low-communication local hidden-variable model exists, or because no single noncontextual hidden state can explain all compatible measurement contexts. This manuscript studies a third regime: genuine global Kochen-Specker contextuality, where local subsystems are noncontextual and the tested multipartite blocks are generalized-Bell-local, but the whole empirical model admits no global noncontextual hidden-variable explanation. We propose a coordination-cost framework in which communication, memory, and local computation are treated as different ways for a classical simulator to maintain a global classical explanation from locally available information. We introduce coordination bits, global contextual covering numbers, scaling laws for task families, and an abstract lifting theorem showing how classical simulation lower bounds for KS-contextual seed families can be transferred to genuinely global-KS models. As worked examples, we analyze a polarization-path Hardy obstruction and postselected KCBS-type tasks.

• The paper introduces the concept of classical coordination cost to quantify the resource gap in simulating genuine global KS contextuality.
• It formalizes a classical cost region (communication, memory, computation) and derives lower bounds using global contextual covering numbers.
• The study bridges Bell nonlocality and KS contextuality, highlighting trade-offs and scalability in distributed classical simulations.

Genuine Global Kochen-Specker Contextuality as a Coordination Cost: A Detailed Analysis

Introduction and Conceptual Framework

This paper establishes a unified operational cost framework for quantum phenomena traditionally analyzed under Bell nonlocality (communication complexity) and Kochen-Specker (KS) contextuality (memory cost), introducing genuine global KS contextuality as a distinct regime. In this regime, all local subsystems are noncontextual and multipartite blocks admit a generalized-Bell-local (GLHV) description, yet the empirical model lacks a global noncontextual explanation. The classical cost of simulating such quantum correlations is formalized as a coordination cost: the minimal sum of communication, memory, and local computation resources (measured in bits) that a classical protocol must expend to maintain global consistency across all contexts.

The author advances the thesis that the operational distinctions between communication, memory, and computation are manifestations of the same underlying informational bottleneck: the need to coordinate local choices into a global classical behavior consistent with quantum empirical models.

Formalization of Coordination Cost

To quantify this classical burden, the notion of a classical cost region is introduced, characterized by the tuple $(B, M, D)$ representing communication (bits), memory (bits), and local computation depth, respectively. The global contextual covering number $X_G(T)$ for a relation task $T$ is defined as the minimum number of global classical assignments (charts) required so that for each context, at least one assignment reproduces the required local statistics. The core lower bound is given by:

$B + M \geq \log_2 X_G(T)$

for any cover-admissible classical protocol. This encapsulates previously distinct communication and memory limitations as projections of a more general coordination constraint.

The cost region thus unifies Bell and KS scenarios: for Bell-type scenarios, the obstruction manifests as necessary communication; for KS, as required internal memory; and for genuine global-KS scenarios, as a coordination resource that cannot be resolved by local assignments, block-local explanations, or their composition.

Depth Restriction and Trade-offs

The framework accommodates limitations on local computation by defining a depth-restricted covering number $x^D(T)$, the minimal number of charts computable by local algorithms of depth $D$ that collectively cover all pertinent contexts. The derived lower bounds show explicit trade-offs:

$D \geq \max\left\{ 0, \log_2 n - (B + M) \right\}$

in a combinatorial model, demonstrating that each additional coordination bit can save one step in local computation depth, but not more.

Example Tasks and Lower Bounds

Twisted Hypercube Parity Task

A toy model, the twisted hypercube, demonstrates that tasks with no global solution require a number of charts (hence coordination bits) that grows with problem size and/or local resource constraints. Explicit calculations show how perfect classical simulation demands increasing coordination complexity as system size grows or as local computation is restricted.

Genuine Global-KS Example: Hardy Obstruction

The central worked example applies the framework to a genuinely global-KS model based on the 2x4 polarization-path Hardy scenario (cf. (Yang, 27 May 2026)), whose subsystems are noncontextual and blocks are GLHV, but which is globally KS-contextual. The associated seven-test relation game is analyzed:

• No deterministic global classical chart wins all tests (maximum single-assignment success: $6/7$).
• Quantum success exceeds classical ($w_Q = (6 + q)/7$ with $q > 0$).
• Thus, for $X_G(T)$0 iterations, a classical simulation matching the quantum rate requires

$X_G(T)$1

This represents a strict classical coordination cost induced purely by genuine global-KS contextuality, not Bell nonlocality or subsystem contextuality.

Strength and Scaling

For postselected KCBS-type constructions, stronger separations (e.g., per-copy rate $X_G(T)$2) can be witnessed, but with additional assumptions (e.g., postselection), whereas unconditional rates from genuine global-KS obstructions remain modest (e.g., $X_G(T)$3 in the Hardy scenario).

Lifting Constructions and Strong Separations

The paper introduces an abstract lifting theorem: any KS-contextual seed model with a strong classical simulation lower bound can be embedded into a genuinely global-KS task (via explicit flag construction) where the lower bound on classical cost is inherited. This shows that strong classical resource separations known for stabilizer subtheories or similar seeds can, in principle, be transferred to the coordination cost for global-KS scenarios, contingent on appropriate model alignment and physical realization.

Practical and Theoretical Implications

The coordination-cost paradigm provides:

• A unified explanation of the classical simulation burden for quantum nonlocality and contextuality.
• A tool for lower-bounding classical resources needed in distributed, memory-restricted, or communication-limited simulations.
• Scalability criteria: Any exponential (or higher) scaling in the global covering number $X_G(T)$4 translates only to linear (or at best quadratic/exponential) scaling in bits. Thus, achieving super-linear separations in practical settings requires task families with rapidly growing $X_G(T)$5, $X_G(T)$6, $X_G(T)$7, or combinatorial rectangle covers, not mere hardness in classical proof systems.
• A bridge to constraint satisfaction and network scenarios, allowing reinterpretation of quantum obstructions in terms of distributed classical database or CSP consistency.

In a broader context, the framework suggests analogous questions for quantum networks and generative models. In quantum networks, the network-level coordination cost measures nonclassicality beyond subsystem or small-block obstructions. In generative modeling, structural analogues of global contextuality can imply irreducible classical sample-coordination complexity.

Conclusion

This work formalizes genuine global Kochen-Specker contextuality as a quantifiable coordination cost for classical simulation, unifying communication, memory, and computation constraints under a common framework. It operationalizes the inability to glue local or blockwise classical explanations into a global one as a concrete bit-based lower bound for classical protocols. While the presented examples verify non-trivial lower bounds, quantitative separation between quantum and classical coordination costs in realistic settings remains modest; stronger rates require construction of task families with rapidly growing global contextual covering numbers or connections to well-characterized quantum resources such as stabilizer magic. The results chart a general direction for future complexity-theoretic and information-theoretic analysis of quantum-classical simulation gaps in distributed, memory-limited, or networked scenarios.

Reference: "Genuine Global Kochen-Specker Contextuality as Classical Coordination Cost" (2606.23577)