Intuitionistic quantum logic of an n-level system

Abstract

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Doering and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (see arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra of complex n by n matrices. This leads to an explicit expression for the pointfree quantum phase space and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen--Specker Theorem. The essential point is that the logical structure of a quantum n-level system turns out to be intuitionistic, which means that it is distributive but fails to satisfy the law of the excluded middle (both in opposition to the usual quantum logic).

26 pages

References in corpus (8)

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• A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
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• Between classical and quantum
• Integrals and Valuations
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Cited by in corpus (20)

• A topos for algebraic quantum theory
• The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach
• Bohrification of operator algebras and quantum logic
• Classical and Quantum Probabilities as Truth Values
• The many classical faces of quantum structures
• Exotic smooth R^4, noncommutative algebras and quantization
• Quantum Mechanics: From Realism to Intuitionism
• The Born rule as structure of spectral bundles (extended abstract)
• Speakable in Quantum Mechanics
• Topos Models for Physics and Topos Theory
• Topos Quantum Logic and Mixed States
• Bohrification: From classical concepts to commutative algebras
• The space of measurement outcomes as a spectrum for non-commutative algebras
• Gelfand spectra in Grothendieck toposes using geometric mathematics
• Intuitionistic interpretation of quantum mechanics
• $L^p$-Wasserstein distances on state and quasi-state spaces of $C^{\ast }$-algebras
• Do quantum propositions obey the principle of excluded middle?
• Equal cost of computation for truth and falsity of experimental quantum propositions necessitates quantum parallel computing
• Speakable in quantum mechanics: babbling on
• Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory