Quantum Modal Logic

Origin. Various approaches combining modal logic and quantum mechanics (2000s-present). Baltag, Smets, Dishkant, Goldblatt. Modal interpretation of quantum observables. Reasoning about quantum measurements and information. Foundation for quantum epistemics.

Models. Modal logic for quantum states. Classical modal: Kripke frames with accessibility. Quantum modal: Hilbert spaces with subspace structure. □φ: necessarily φ (eigenstate of observable). Dynamic: measurement updates knowledge.

Formalism.

Static quantum modal logic: Propositions = closed subspaces of Hilbert space H. a ⊨ P iff state a ∈ P (subspace).

Modal operator via observable A: □_A φ: "measuring A would definitely give result compatible with φ" State in eigenspace of A corresponding to values in ⟦φ⟧.

Dynamic quantum modal logic: [M]φ: after measurement M, φ holds. Measurement updates state (projection).

Semantics: ⟦p⟧ = closed subspace (quantum proposition) ⟦□φ⟧ = {ψ : ψ in eigenspace implies ψ ∈ ⟦φ⟧} ⟦[M]φ⟧ = {ψ : Proj_M(ψ) ∈ ⟦φ⟧}

Quantum epistemic logic: Agent's knowledge = compatible states. Kφ: agent knows φ (all compatible states satisfy φ). Measurement reveals information.

Orthomodular modal logic: Non-distributive: □(φ ∧ ψ) ≠ □φ ∧ □ψ in general. Reflects quantum superposition.

Symbols.

Symbol	Unicode	Name	Meaning
□	U+25A1	Necessity	Observable certainty
◇	U+25C7	Possibility	Compatible
[M]	—	Measurement	After measuring M
⟦φ⟧	—	Interpretation	Subspace
K	—	Knows	Epistemic
⊥	U+22A5	Orthogonal	Perpendicular
Proj	—	Projection	Measurement

Metatheory. Quantum modal logic non-classical. Orthomodular but not Boolean. Decidability: depends on fragment. Complete for quantum frame semantics. Connects to quantum logic tradition. Measurement dynamics well-founded.

Applies to. Quantum information theory. Quantum protocols. Quantum computing foundations. Philosophy of quantum mechanics. Quantum game theory. Epistemic quantum states.

Limitations. Non-classical reasoning required. Multiple incompatible frameworks. Connection to actual quantum computation indirect. Tool support minimal. Theoretical focus. Learning curve from both modalities.