Contextuality as an External Bookkeeping Cost under Fixed Shared-State Semantics
Summary: arXiv:2601.20167v2 Announce Type: cross
Abstract
Contextuality is a central feature distinguishing quantum from classical probability theories, but its operational meaning is often stated only qualitatively. In this Letter, we study a simple information-theoretic question: how much additional contextual information must a classical simulation introduce when it tries to keep a shared internal description fixed across contexts? To make this question precise, we analyze a minimal external-label simulation model in which the remaining context dependence is carried only by an auxiliary label. For this model, we define an obstruction cost as the minimum mutual information between the context and the auxiliary label required to reproduce the observed statistics.
Key Findings
We then prove a conservative quantitative lower bound: any linear witness that separates the observed statistics from the zero-obstruction set yields a positive lower bound on this cost. We do not claim that this bound is tight, and we do not claim that the simulation model covers every possible classical architecture. Its role is narrower and more explicit: under fixed shared-state semantics, contextuality can be read as a certificate of irreducible external bookkeeping cost in a simple and well-defined simulation model.
Understanding Contextuality
Contextuality refers to the phenomena where the outcome of a measurement cannot be understood independently of the context of other measurements. This characteristic is crucial in differentiating quantum systems from classical frameworks. The operational implications of contextuality, however, are often left ambiguous, leading to a need for a clearer understanding.
Methodology
In our approach, we focus on a minimal simulation model designed to analyze contextuality quantitatively. The model operates under fixed shared-state semantics, where the context of an experiment is represented by an auxiliary label. This label serves as a bridge to explore how much contextual information is necessary for a classical simulation to reproduce observed quantum statistics.
Defining Obstruction Cost
The obstruction cost becomes a pivotal concept in our analysis. It quantifies the minimum amount of mutual information that must exist between the context and the auxiliary label for accurate statistical reproduction. By establishing this cost, we can assess the extent to which contextuality influences the capabilities of classical simulations.
Results and Implications
- We demonstrate that any linear witness that effectively distinguishes the observed statistics from the zero-obstruction set indicates a non-zero obstruction cost.
- This finding has implications for understanding the limitations of classical models in replicating quantum phenomena.
- While we acknowledge that our bounds may not be definitive, they provide a foundational framework for further exploration into contextuality and its implications in quantum mechanics.
Conclusions
In summary, our analysis elucidates the role of contextuality as a fundamental aspect of quantum mechanics that carries significant operational meaning. By framing contextuality within the context of external bookkeeping costs, we open the door to a deeper understanding of the relationship between quantum and classical systems. Our findings pave the way for future research, potentially leading to advancements in quantum information theory and computation.