From Boolean contexts to quantum logic: A left adjoint construction and its biological implications

Area of Science:

• Quantum Information Theory
• Category Theory
• Lattice Theory

Background:

• Contextuality is a key feature of quantum information.
• Its origins are often unclear due to probabilistic formulations.
• This work reconstructs contextuality from classical contexts.

Purpose of the Study:

• To demonstrate that non-distributive orthomodular structure arises from classical Boolean contexts.
• To provide a categorical framework for understanding quantum logic.
• To establish a link between contextuality, non-distributivity, and sheaf theory.

Main Methods:

• Introduced a gluing functor using categorical pushouts on Boolean algebras.
• Established a left adjoint relationship between the constructed lattice and Boolean subalgebras.
• Utilized sheaf theory to analyze the failure of Boolean properties.

Main Results:

• The gluing construction yields orthomodular, non-distributive lattices.
• This construction is left adjoint to a forgetful functor, generating quantum logic from classical contexts.
• Failure of Boolean properties corresponds to the absence of global sections in sheaf theory.

Conclusions:

• Quantum contextuality and non-distributivity can be structurally derived from classical contexts.
• This provides a categorical and lattice-theoretic foundation for quantum logic.
• The gluing construction offers a model for emergent global coherence in systems with local classical contexts.