Quantum Foundations / Information Theory
Contextuality from Single-State Ontological Models: An Information-Theoretic Obstruction
Contextuality in quantum theory, traditionally understood as the impossibility of reproducing quantum measurement statistics with noncontextual ontological models, is re-examined. This paper introduces an information-theoretic obstruction within classical single-state ontological descriptions: whenever such a model reproduces operational statistics using an auxiliary contextual register, the required contextual information is lower-bounded by the conditional mutual information I(C; O | λ). This highlights that under shared-state reuse, contextual distinctions may not be fully internalized within the subsystem ontic state alone, framing contextuality as a representational limitation rather than a dualism about physical reality.
Executive Impact Metrics
The paper quantifies the information cost of representing contextual distinctions in classical models reusing a single ontic state. This cost is directly tied to the conditional mutual information.
Deep Analysis & Enterprise Applications
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Information Theory
Quantum Foundations
I(C; O | λ)
Information-Theoretic Obstruction Defined
The core result is an information-theoretic obstruction: if a classical single-state ontological model reproduces operational statistics using an auxiliary contextual register M, the required contextual information is lower-bounded by the conditional mutual information I(C; O | λ). This implies that H(M) > I(C; O | λ), where C is the intervention, O is the outcome, and λ is the ontic state.
Enterprise Process Flow
Operational Statistics
→
Common Ontic State (λ) Reuse
→
Auxiliary Contextual Register (M)
→
Contextual Dependence (I(C;O|λ))
→
Required H(M) > 0
Constructive Illustration: Toy Model
Summary: A simple deterministic toy example illustrates the I(C; O | λ) > 0 regime. If outcome O = λ + f(C), where f(C) is an intervention-dependent bit and λ, C are independent, then I(λ; O) = 0 but I(C; O | λ) > 0. This implies that even with full knowledge of the ontic state λ, intervention C still provides information about O, directly requiring H(M) > 0 for an auxiliary register.
Enterprise Impact: Demonstrates how contextual dependence requiring auxiliary bookkeeping can arise operationally under single-state constraints, without assuming quantum dynamics or Hilbert space structure.
The significance of the bound is interpretive: it highlights a 'bookkeeping cost' when intervention-dependent distinctions are absorbed into an auxiliary contextual register, rather than being fully internalized within the subsystem ontic state. This is a representational constraint, not an inherent physical limitation.
| Feature | Classical Single-State Model (with M) | Quantum Theory |
|---|---|---|
| Ontic State Space | Fixed, reused across interventions, not indexed by context | Single quantum state ρ reused, but not assumed to define a single underlying classical variable for all contexts |
| Contextual Dependence | Mediated by response functions and auxiliary register M; I(C;O|λ) > 0 requires H(M) > 0 | Handled by Born rule; not forced into a single global classical probability space without refinement |
| Origin of Constraint | Representational constraint: single global classical probability space over ontic states | Structural difference; avoids the classical constraint |
| Bookkeeping Cost | Explicit cost H(M) tied to I(C;O|λ) | No direct equivalent of M or I(C;O|λ) in the same way |
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AI Implementation Roadmap
The research suggests several future directions, including extending the analysis to dynamical settings, characterizing minimal contextual information costs, and exploring connections to information complexity and representational efficiency.
Phase 1: Extend Analysis to Dynamical Settings
Investigate how the information-theoretic obstruction behaves in evolving systems and time-dependent interventions.
Phase 2: Characterize Minimal Contextual Information Costs
Quantify the minimum H(M) required for specific contextuality scenarios beyond the general lower bound.
Phase 3: Explore Information Complexity and Efficiency
Connect the findings to broader theories of information complexity and representational efficiency in classical and nonclassical models.
Phase 4: Generalize to Adaptive Systems
Apply the framework to resource-limited adaptive and intelligent systems to understand contextuality as a representational constraint.
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