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 Döring 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 (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra M n (C) of complex n × n matrices. This leads to an explicit expression for the pointfree quantum phase space n 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 KochenSpecker Theorem.In our approach, the nondistributive lattice P(M n (C)) of projections in M n (C) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice O( n ) of functions from the poset C(M n (C)) of all unital commutative C*-subalgebras C of M n (C) to P(M n (C)). The lattice O( n ) is essentially the (pointfree) topology of the quantum phase space n , Dedicated to Pekka Lahti, at his 60th birthday.M. Caspers ( ) · C. Heunen · N.P. Landsman 732 Found Phys (2009) 39: 731-759 and as such defines a Heyting algebra. Each element of O( n ) corresponds to a "Bohrified" proposition, in the sense that to each classical context C ∈ C(M n (C)) it associates a yes-no question (i.e. an element of the Boolean lattice P(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.Keywords Quantum logic · Topos theory · Intuitionistic logic 'All departures from common language and ordinary logic are entirely avoided by reserving the word "phenomenon" solely for reference to unambiguously communicable information, in the account of which the word "measurement" is used in its plain meaning of standardized comparison. ' (N. Bohr [4])

The Many Classical Faces of Quantum Structures

“…• Either one uses C(A); the canonical contextual set C → C is a commutative C*-algebra, and the spectral contextual set X does not take a canonical form [72,73,21,74,126,109]. • Or one uses the opposite order; the spectral contextual set X is a locale of the canonical form C → Spec(C), and the commutative C*-algebra C(X) does not take a canonical form [40,41,39,49].…”

“…We can only touch on it briefly here, but one of the main features of building the topos of contextual sets over C(A) and distilling the configuration space Spec(A) is that they encode a contextual logic. This logic is intuitionistic, and therefore very different from traditional quantum logic [21]. The latter concerns the set Proj(A) of yes-no questions on the quantum system A; more precisely, the set of sharp observables with two outcomes.…”

Uncertainty Relations and Possible Experience

“…Within the topos approach begun by Chris Isham and Jeremy Butterfield and continued with Andreas Doering and others, it was noted that there is an associated intuitionist logic that can be used that has the potential to avoid difficulties encountered in the quantum logical approach, one considering the formal language of quantum theory via the notion of Boolean contexts [36,37]. A topos is a category that can be considered a generalized universe of sets with an internal intuitionist logic offering the possibility of distributivity that promises greater structural similarity with classical theory.…”

Constructive Pointfree Topology Eliminates Non-constructive Representation Theorems from Riesz Space Theory

“…In turn, the results about commutative C*-algebras may be obtained directly using Riesz spaces [14]. This is used in applications of topos theory to quantum theory [10,16,17].…”