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Proceedings of the American Mathematical Society

Quantum Logics with Lattice State Spaces

Jiří Binder, Mirko Navara

Published Jan 1, 1987• 5 citations
Publisher

Abstract

Let $L$ be a quantum logic and let $S(L)$ denote the set of all states on $L$. (By a state we mean a nonnegative bounded $\sigma$-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].

DOI
10.1090/s0002-9939-1987-0894439-1

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Let LLL be a quantum logic and let S(L)S(L)S(L) denote the set of all states on LLL. (By a state we mean a nonnegative bounded σ\sigmaσ-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].
It is shown that for any quantum logic LLL one can find a concrete logic KKK and a surjective homomorphism fff from KKK onto LLL such that fff maps the centre of KKK onto the centre of LLL. Moreover, one can ensure that each finite set of compatible elements in LLL is the image of a compatible subset of KKK. This result is "best possible" - let a logic LLL be the homomorphic image of a concrete logic under a homomorphism such that, if FFF is a finite subset of the pre-image of a compatible subset of LLL, then FFF is compatible. Then LLL must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.
Recently we proved that there are two non-isomorphic models of the calculus of quantum logic corresponding to an infinite-dimensional Hilbert space representation: an orthomodular lattice and a weakly orthomodular lattice. We also discovered that there are two non-isomorphic models of the calculus of classical logic: a distributive lattice (Boolean algebra) and a weakly distributive lattice. In this work we consider implications of these results to a quantum simulator which should mimic quantum systems by giving precise instructions on how to produce input state, how to evolve them, and how to read off the final states. We analyze which conditions quantum states of a quantum computer currently obey and which they should obey in order to enable full quantum computing, i.e., proper quantum mathematics. In particular we find several new conditions which lattices of Hilbert space subspaces must satisfy.
A set of statements about the properties of a quantum system is looked at as at a partially ordered set of subspaces of finite or infinite dimensional Hilbert space. The operation of ordering is performed on a set of propositions comparing the truth values of these propositions and on the set of subspaces as the opera- tion of inclusion. Based on the required properties these structures are translated into operations on the lattice. The correspondence with Heisenberg uncertainty principle is shown there. Furthermore, it is shown that the lattices correspond- ing to the subspaces of infinite dimensional Hilbert space are not modular. This property is replaced with weaker property of orthomodularity, when operation of the negation is added. Following the work of G. Birkhoff and J. von Neumann, the structure of quantum logic is looked for in projective spaces, which are in- troduced either arithmetically or axiomatically. The examples of quantum logic, their physical implementation and eventual implementation in projective spaces are analysed. 1
Following Birkhoff and von Neumann, quantum logic has traditionally been based\non the lattice of closed linear subspaces of some Hilbert space, or, more generally,\non the lattice of projections in a von Neumann algebra A. Unfortunately, the logical\ninterpretation of these lattices is impaired by their nondistributivity and by various\nother problems. We show that a possible resolution of these difficulties, suggested by\nthe ideas of Bohr, emerges if instead of single projections one considers elementary\npropositions to be families of projections indexed by a partially ordered set C(A) of\nappropriate commutative subalgebras of A. In fact, to achieve both maximal generality\nand ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra\nand that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such\nfamilies of projections form a Heyting algebra in a natural way, so that the associated\npropositional logic is intuitionistic: distributivity is recovered at the expense of the\nlaw of the excluded middle.\nSubsequently, generalizing an earlier computation for n×n matrices, we prove that\nthe Heyting algebra thus associated to A arises as a basis for the internal Gelfand\nspectrum (in the sense of Banaschewski–Mulvey) of the "Bohrification" A of A, which\nis a commutative Rickart C*-algebra in the topos of functors from C(A) to the category\nof sets. We explain the relationship of this construction to partial Boolean algebras\nand Bruns–Lakser completions. Finally, we establish a connection between probability\nmeasure on the lattice of projections on a Hilbert space H and probability valuations\non the internal Gelfand spectrum of A for A = B(H).
Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A){\mathcal{C}(A)}C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A){\mathcal{C}(A)}C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” A‾{\underline A}A​ of A, which is a commutative Rickart C*-algebra in the topos of functors from CA{\mathcal{C}A}CA to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A‾{\underline A}A​ for A = B(H).