It is well known that the “quantum logic” approach to the foundations of quantum mechanics is based on the subspace ordering of projectors on a Hilbert space. In this paper, we show that this is a special case of an ordering on partial isometries, introduced by Halmos and McLaughlin. Partial isometries have a natural physical interpretation, however, they are notoriously not closed under composition. In order to take a categorical approach, we demonstrate that the Halmos-McLaughlin partial ordering, together with tools from both categorical logic and inverse categories, allows us to form a category of partial isometries. This category can reasonably be considered a “categorification” of quantum logic – we therefore compare this category with Abramsky and Coecke's “compact closed categories” approach to foundations and with the “monoidal closed categories” view of categorical logic. This comparison illustrates a fundamental incompatibility between these two distinct approaches to the foundations of quantum mechanics. 9.1 Introduction As early as 1936, von Neumann and Birkhoff proposed treating projectors on Hilbert space as propositions about quantum systems (Birkhoff and von Neumann 1936), by direct analogy with classical order-theoretic approaches to logic. Boolean lattices arise as the Lindenbaum-Tarski algebras of propositional logics, and as the set of all projectors on a Hilbert space also forms an orthocomplemented lattice, the operations meet, join , and complement were analogously interpreted as the logical connectives conjunction, disjunction , and negation .
It is well known that the “quantum logic” approach to the foundations of quantum mechanics is based on the subspace ordering of projectors on a Hilbert space. In this paper, we show that this is a special case of an ordering on partial isometries, introduced by Halmos and McLaughlin. Partial isometries have a natural physical interpretation, however, they are notoriously not closed under composition. In order to take a categorical approach, we demonstrate that the Halmos-McLaughlin partial ordering, together with tools from both categorical logic and inverse categories, allows us to form a category of partial isometries. This category can reasonably be considered a “categorification” of quantum logic – we therefore compare this category with Abramsky and Coecke's “compact closed categories” approach to foundations and with the “monoidal closed categories” view of categorical logic. This comparison illustrates a fundamental incompatibility between these two distinct approaches to the foundations of quantum mechanics. 9.1 Introduction As early as 1936, von Neumann and Birkhoff proposed treating projectors on Hilbert space as propositions about quantum systems (Birkhoff and von Neumann 1936), by direct analogy with classical order-theoretic approaches to logic. Boolean lattices arise as the Lindenbaum-Tarski algebras of propositional logics, and as the set of all projectors on a Hilbert space also forms an orthocomplemented lattice, the operations meet, join , and complement were analogously interpreted as the logical connectives conjunction, disjunction , and negation .
Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the ‘quantum logic’ of subspaces of a general vector space—which is then specialized to the closed subspaces of a Hilbert space. But there is a ‘dual’ progression. The set notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions (i.e. the vector space version of a set partition) of a general vector space—which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets.
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 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) 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) 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 of A, which is a commutative Rickart C*-algebra in the topos of functors from 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 for A = B(H).
Elaborating on our joint work with Abramsky in [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124] we further unravel the linear structure of Hilbert spaces into several constituents. Some prove to be very crucial for particular features of quantum theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases. First we show that the bulk of the required linear structure is purely multiplicative, and arises from the strongly compact closed tensor which, besides providing a variety of notions such as scalars, trace, unitarity, self-adjointness and bipartite projectors [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124], also provides Hilbert-Schmidt norm, Hilbert-Schmidt inner-product, and in particular, the preparation-state agreement axiom which enables the passage from a formalism of the vector space kind to a rather projective one, as it was intended in the (in)famous Birkhoff & von Neumann paper [G. Birkhoff, J. von Neumann, (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823–843]. Next we consider additive types which distribute over the tensor, from which measurements can be build, and the correctness proofs of the protocols discussed in [S. Abramsky, B. Coecke, (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130] carry over to the resulting weaker setting. A full probabilistic calculus is obtained when the trace is moreover linear and satisfies the diagonal axiom, which brings us to a second main result, characterization of the necessary and sufficient additive structure of a both qualitatively and quantitatively effective categorical quantum formalism without redundant global phases. Along the way we show that if in a category a (additive) monoidal tensor distributes over a strongly compact closed tensor, then this category is always enriched in commutative monoids.
According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.
Quantum Logic Distributivity And The Algebraic Structure Of Propositions As with many subatomic particles, the question of under what conditions the electron behaves as a particle or as a wave continues to be the subject of research. It was necessary to employ appropriate logic to organise the propositions describing the behaviour of such an object at the subatomic scale into a precise from and to establish coherent relationships between them. Quantum logic, which can be regarded as an interpretation of quantum mechanics, arose from this need. In classical mechanics, the state and properties of a system are described using the propositions of classical logic, which is based on Boolean algebra. In quantum mechanics, however, the fact that the operators representing the measured quantities do not commute has shown that quantum phenomena cannot be described using the propositions of classical logic. It is therefore necessary to have a formal framework suited to the propositions describing quantum phenomena. In our study, we highlight the various differences between classical logic and quantum logic, examine the distributive property and discuss, from the perspective of Birkhoff and von Neumann, the algebraic structure to which quantum logic might correspond. At the conclusion of our research, we find that, in order to assert that all experimental propositions logically equivalent to a given experimental proposition constitute a physical property, the operators corresponding to the measured quantities must commute with one another. Boolean algebra has an algebraic structure that allows classical logic to be formalised. In such algebra, the distributive property holds; thus, distributivity does not pose a problem for classical logic. Consequently, the distributive property does not always hold in quantum logic and quantum systems can be modelled using orthomodular lattices associated with closed subspaces of Hilbert space. We believe that this model must evolve in parallel with the discovery of new quantum phenomena and that it provides a framework conductive to this evolution. Keywords: Quantum logic; distributivity; partially order sets (posets); lattice; orthomodular lattice; Hilbert space; Boolean algebra.