Abstract. In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice Lsep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms ” of the lab, and before any interaction takes place. In that case the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore Lsep cannot satisfy all Piron’s axioms. In previous works, assuming that Lsep is orthocomplemented, it was argued that Lsep is not orthomodular and fails to have the covering property. Here we prove that Lsep cannot admit and orthocomplementation. Moreover, we propose a natural model for Lsep which has the covering property. PACS numbers: 03.65.Ta, 03.65.Ca 1.
Abstract. In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice Lsep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms ” of the lab, and before any interaction takes place. In that case the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore Lsep cannot satisfy all Piron’s axioms. In previous works, assuming that Lsep is orthocomplemented, it was argued that Lsep is not orthomodular and fails to have the covering property. Here we prove that Lsep cannot admit and orthocomplementation. Moreover, we propose a natural model for Lsep which has the covering property. PACS numbers: 03.65.Ta, 03.65.Ca 1.
Abstract. Given two complete atomistic lattices L1 and L2, we define a set S = S(L1, L2) of complete atomistic lattices by means of three axioms (natural regarding the description of quantum compound systems), or in terms of a universal property with respect to a given class of bimorphisms. We prove that S is a complete lattice. The bottom element L1 ∧○L2 is the separated product of Aerts. For atomistic lattices with 1 (not complete), L1 ∧○L2 ∼ = L1□L2 the box product of Grätzer and Wehrung, and, in case L1 and L2 are moreover coatomistic, L1 ∧○L2 ∼ = L1 ⊠ L2 the lattice tensor product. The top element L1 ∨○L2 is the (complete) join-semilattice tensor product of Fraser, which is isomorphic to the tensor products of Chu and Shmuely. With some additional hypotheses on L1 and L2 (true if L1 and L2 are moreover orthomodular with the covering property), we prove that S is a singleton if and only if L1 or L2 is distributive, if and only if L1 ∨○L2 has the covering property. Our main result reads: L ∈ S admits an orthocomplementation if and only if L = L1 ∧○L2. For L1 and L2 moreover irreducible, we characterize the automorphisms of each L ∈ S in terms of those of L1 and L2. At the end, we construct an example L1 ⇓○L2 in S which has the covering property. 1.
It has been shown by Birkhoff and v. Neumann (1936) and by Jauch and Piron (1963,1964,1968) that the subspaces of Hilbert space constitute an orthocomplemented quasi-modular lattice Lq, if one considers between two subspaces (elements) a, b the relation a⊆b and the operations a∩b, a∪b, a⊥. Furthermore, since the subspaces can be interpreted as quantum mechanical propositions, and since the operations ∩,∪ ,⊥ have some similarity with the logical operations ⋀ (and), ⋁ (or) and ⌝ (not), the question has been raised already by Birkhoff and v. Neumann, whether the lattice of subspaces of Hilbert space can be interpreted as a propositional calculus, sometimes called quantum logic. There are many kinds of lattices which can be interpreted as a propositional or logical calculus. A Boolean lattice LB of propositions corresponds to the calculus of classical logic and an implicative (Birkhoff, 1961) lattice Li has as a model the calculus of effective (intuitionistic) logic.
Two new possibilities in the axiomatic foundations of quantum mechanics are examined: first, the possibility of introducing a non-symmetric transition probability between pure states, and second, showing that the concept of orthocomplementation in the logic of events is unnecessary and of secondary importance. An axiomatic scheme is presented which does not involve the concept of orthocomplementation and yet has all the advantages of the well-known quantum logic axiomatics, because the generalised logic of events admits an extension, which is a complete orthocomplemented orthomodular lattice with the covering law holding in it. Thus, the approach to quantum axiomatics may be seen as answering both the old questions of the quantum logic approach (e.g. the questions of the complete lattice structure of the logic, atomicity, the validity of the covering law) and the question concerning the necessity of the orthocomplementation in the logic of events, recently raised by Mielnik (1976).
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.
We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects that are considered, and mathematically described, as one compound system. It is shown that the separated product of the property lattices corresponding with the two spin 1/2 objects does not have an underlying linear structure, although the property lattices associated with the subobjects in isolation manifestly do. This is related at a fundamental level to the fact that separated products do not behave well with respect to the covering law (and orthomodularity) of elementary lattice theory. In addition, we discuss the orthogonality relation associated with the separated product in general and consider the related problem of the behavior of the corresponding Sasaki projections as partial state space mappings
We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects that are considered, and mathematically described, as one compound system. It is shown that the separated product of the property lattices corresponding with the two spin 1/2 objects does not have an underlying linear structure, although the property lattices associated with the subobjects in isolation manifestly do. This is related at a fundamental level to the fact that separated products do not behave well with respect to the covering law (and orthomodularity) of elementary lattice theory. In addition, we discuss the orthogonality relation associated with the separated product in general and consider the related problem of the behavior of the corresponding Sasaki projections as partial state space mappings