# Distributivity breaking and macroscopic quantum games

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Quantum logic can be understood in two ways: as a study of the algebraic structures that appear in the context of the Hilbert space formalism of quantum mechanics; or as representing a non-classical logic in conflict with classical logic. My aim in this paper is to analyze the possibility to...

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The complete orthomodular lattice of closed subspaces of a Hilbert space is considered as the logic describing a quantum physical system, and called a quantum logic. G. Takeuti developed a quantum set theory based on the quantum logic. He showed that the real numbers defined in the quantum set...

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The original proof of Gleasonâ€™s Theorem is very complicated and therefore, any result that can be derived also without the use of Gleasonâ€™s Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved...

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In 2006, Gudder introduced a logic order on the bounded quantum observable set S(H). In 2007, Pulmannova and Vincekova proved that for each subset D of S(H), the infimum of D exists with respect to this logic order. In this paper, we present a representation theorem for the infimum of D.

- Quantum logic as a dynamic logic. Baltag, Alexandru; Smets, Sonja // Synthese;Mar2011, Vol. 179 Issue 2, p285
We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear 'no'....

- Gentzen-Type Calculi for Involutive Quantales. Kamide, Norihiro // International Journal of Theoretical Physics;Apr2005, Vol. 44 Issue 4, p399
Completeness and cut-elimination theorems are proved for some Gentzen-type sequent calculi which are closely related to non-commutative involutive quantales.

- Product Åukasiewicz Quantum Logic. Bertini, Cesarino; Leporini, Roberto // International Journal of Theoretical Physics;Feb2011, Vol. 50 Issue 2, p571
The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a density operator (called qumix). In this framework, any sentence Î± of the...

- Quantum Computational Logic. Gudder, S. // International Journal of Theoretical Physics;Jan2003, Vol. 42 Issue 1, p39
A quantum computational logic is constructed by employing density operators on spaces of qubits and quantum gates represented by unitary operators. It is shown that this quantum computational logic is isomorphic to the basic sequential effect algebra [0, 1].

- Typed Quantum Logic. Tokuo, Kenji // International Journal of Theoretical Physics;Jan2003, Vol. 42 Issue 1, p27
The aim of this paper was to lift traditional quantum logic to its higher order version with the help of a type-theoretic method. A higher order axiomatic system is defined explicitly and then a sound and complete class of models is given. This is an attempt to provide a quantum counterpart of...