Quantum modal logic
Kenji Tokuo
TL;DR
The paper introduces Quantum Modal Logic (QML), a framework that integrates modal operators with Birkhoff–von Neumann quantum logic by combining relational semantics for quantum logic with Kripke-style modal semantics. It defines a quantum modal structure $\langle W,R_Q,R_M,\rho\rangle$, where $R_Q$ is reflexive and symmetric and $R_M$ is forced by $R_Q$, and it provides a sequent-calculus axiomatization that extends Nishimura's non-modal quantum logic with MEM and K. The authors prove soundness and completeness of QML by constructing a canonical model and showing the truth of formulas corresponds to derivability, and they illustrate how different choices of $R_M$ realize alethic, temporal, and dynamic quantum logics. The work offers a basis for formalizing various quantum logics (e.g., quantum temporal, epistemic, dynamic) and discusses future work on relaxing forcing constraints for improved applicability. The results lay groundwork for rigorous specification and reasoning about modal phenomena in quantum settings, with potential impact on quantum computing, quantum information, and foundations of quantum theory.
Abstract
A modal logic based on quantum logic is formalized in its simplest possible form. Specifically, a relational semantics and a sequent calculus are provided, and the soundness and the completeness theorems connecting both notions are demonstrated. This framework is intended to serve as a basis for formalizing various modal logics over quantum logic, such as quantum alethic logic, quantum temporal logic, quantum epistemic logic, and quantum dynamic logic.