The quantum propositional logic of Birkhoff and von Neumann is the logic of Boolean observables on a quantum system. The quantum predicate logic of Weaver extends this quantum propositional logic to a logic of Boolean observables on composite quantum systems. We investigate a variant of this quantum predicate logic in which the physical systems are modeled by hereditarily atomic von Neumann algebras. Such physical systems may be infinite-dimensional, but their superselection sectors are finite-dimensional. From the perspective of noncommutative mathematics, they are a quantum generalization of sets. We define an equality relation on each such quantum set. This definition allows us to axiomatize several classes of discrete quantum structures within quantum predicate logic, including quantum graphs, quantum groups and quantum metric spaces. We likewise characterize the quantum isomorphism of simple graphs. The proof that discrete quantum groups may be so axiomatized is due to Stefaan Vaes.
The quantum propositional logic of Birkhoff and von Neumann is the logic of Boolean observables on a quantum system. The quantum predicate logic of Weaver extends this quantum propositional logic to a logic of Boolean observables on composite quantum systems. We investigate a variant of this quantum predicate logic in which the physical systems are modeled by hereditarily atomic von Neumann algebras. Such physical systems may be infinite-dimensional, but their superselection sectors are finite-dimensional. From the perspective of noncommutative mathematics, they are a quantum generalization of sets. We define an equality relation on each such quantum set. This definition allows us to axiomatize several classes of discrete quantum structures within quantum predicate logic, including quantum graphs, quantum groups and quantum metric spaces. We likewise characterize the quantum isomorphism of simple graphs. The proof that discrete quantum groups may be so axiomatized is due to Stefaan Vaes.
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum sets are certain binary relations that can be characterized in terms of this dagger compact structure, and the resulting category of quantum sets and functions generalizes the category of ordinary sets and functions in the manner of noncommutative mathematics. In particular, this category is dual to a subcategory of von Neumann algebras. The basic properties of quantum sets are presented thoroughly, with the noncommutative dictionary in mind, and with an eye to convenient application. As a motivating example, a notion of quantum graph coloring is derived within this framework, and it is shown to be equivalent to the notion that appears in the quantum information theory literature.
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
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum sets are certain binary relations that can be characterized in terms of this dagger compact structure, and the resulting category of quantum sets and functions generalizes the category of ordinary sets and functions in the manner of noncommutative mathematics. In particular, this category is dual to a subcategory of von Neumann algebras. The basic properties of quantum sets are presented thoroughly, with the noncommutative dictionary in mind, and with an eye to convenient application. As a motivating example, a notion of quantum graph coloring is derived within this framework, and it is shown to be equivalent to the notion that appears in the quantum information theory literature.
Sentences Associated with a Quantum System and Their Truth Values In a sense this chapter is marginal to the content of this volume; the reader can skip it without prejudice to the comprehension of chapters to follow. Quantum logic is a discipline that branched off from the 1936 paper of Birkhoff and von Neumann and has the orthomodular-poset structures encountered in previous chapters as basic mathematical carriers: that is why we think it worthwhile to devote some pages to the subject. But the main interests quantum logic calls into play are in the territory of logicians and philosophers: that is why we (who are neither logicians nor philosophers) shall confine ourselves to a very simplified introduction to the subject. Roughly, the starting question is whether the propositions of a quantum system can be associated with, or can be interpreted as, sentences of a language (or propositional calculus) and which rules this language inherits from the ordered structure of propositions. In raising this question one has in mind the fact that when the physical system is classical its propositions form a Boolean algebra, and Boolean algebras are the algebraic models of the calculus of classical logic. Thus the question above can also be phrased as follows: when a Boolean algebra is relaxed into an orthomodular nondistributive lattice, which logic is it the model of? “Quantum logic” is the name that designates the answer, but there are several views about the content of this name. Here we sketch one approach to a quantum logic.
Birkhoff and von Neumann [Ann. Math. 37, 823 (1936)] proposed a nonstandard logic to describe quantum mechanics, in which the distributive laws of Boolean logic do not hold. In this paper we develop two algebras of propositions for classical mechanics that, like the quantum logical algebra, are based on a measurement theory. We adopt a simple classical measurement theory that allows the determination of any continuous phase-space function to any finite precision. Surprisingly, the resulting ``classical logics'' are non-Boolean, though the distributive laws hold. It appears that any physical theory with a mathematical space of physical states and an adequate description of measurement naturally yields a logiclike structure of experimental propositions, and that this ``derived logic'' can be non-Boolean even for theories much less radical than quantum theory.
Due to the failure of the classical principles of bivalence and verifunctionality, the logic of experimental propositions relative to quantum systems cannot be interpreted in Boolean algebras. However, we cannot say neither that this logic is captured by orthomodular lattices, as claimed by many authors along the line of Birkhoff's and von Neumann's standard approach. For the alleged violation of distributivity is based on the possibility of combining statements relative to complementary contexts, which does not refer to any experience and, consequently, has no meaning. Indeed, quantum logic should be interpreted in partial, transitive Boolean algebras whose compatibility relation limits the application of the connectives within each of its Boolean sub-algebras, which refer to partial, classical descriptions. Moreover, this approach of quantum logic makes it possible to deal with composite systems, which was not possible to do within the standard approach, and then to deal with the fundamental notion of quantum entanglement. The latter notion can be represented by a series of axioms of the object language that restrict the set of experimental statements bearing on a composite system, while its close link to the notion of complementarity can be expressed in the metalanguage.