A quantum system Σ( d ) with variables in Z( d ) and with Hilbert space H ( d ), is considered. It is shown that the additivity relation of Kolmogorov probabilities, is not valid in the Birkhoff-von Neumann orthocomplemented modular lattice of subspaces L ( d ). A second lattice Λ( d ) which is distributive and contains the subsystems of Σ( d ) is also considered. It is shown that in this case also, the additivity relation of Kolmogorov probabilities is not valid. This suggests that a more general (than Kolmogorov) probability theory is needed, and here we adopt the Dempster-Shafer probability theory. In both of these lattices, there are sublattices which are Boolean algebras, and within these 'islands' quantum probabilities are additive.
A quantum system Σ( d ) with variables in Z( d ) and with Hilbert space H ( d ), is considered. It is shown that the additivity relation of Kolmogorov probabilities, is not valid in the Birkhoff-von Neumann orthocomplemented modular lattice of subspaces L ( d ). A second lattice Λ( d ) which is distributive and contains the subsystems of Σ( d ) is also considered. It is shown that in this case also, the additivity relation of Kolmogorov probabilities is not valid. This suggests that a more general (than Kolmogorov) probability theory is needed, and here we adopt the Dempster-Shafer probability theory. In both of these lattices, there are sublattices which are Boolean algebras, and within these 'islands' quantum probabilities are additive.
The orthocomplemented modular lattice of subspaces L[H(d)]L[H(d)], of a quantum system with d-dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities is violated by quantum probabilities in the full lattice L[H(d)]L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)]L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2)D(H1,H2), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2)P(H1),P(H2), to the subspaces H1, H2. As an application, it is shown that the proof of the inequalities of Clauser, Horne, Shimony, and Holt for a system of two spin 1/2 particles is valid for Kolmogorov probabilities, but it is not valid for Dempster-Shafer probabilities. The violation of these inequalities in experiments supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.
The set of subsystems Σ m ( ) of a finite quantum system Σ n ( ) (with variables in n ( )) together with logical connectives, is a distributive lattice.With regard to this lattice, the) Tr( ( ) ) n n (where P m ( ) is the projector to Σ m ( )) obeys a supermodularity inequality, and it is interpreted as a lower probability in the sense of the Dempster-Shafer theory, and not as a Kolmogorov probability.It is shown that the basic concepts of the Dempster--Shafer theory (lower and upper probabilities and the Dempster multivaluedness) are pertinent to the quantum formalism of finite systems.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
This thesis is a review of the quantum logic approach to quantum probability theory which was first studied by Birkhoff and von Neumann in 1936. Our study is based on the more general quantum structure which is known as the sigma-orthocomplete orthomodular poset or quantum logic. We introduce the definitions of states and observables and show that the quantum probability theory is a generalization of the classical probability theory due to Kolmogorov. Discussions about joint distributions of observables are given, and we also give the general expression of the uncertainty relation in the logico-algebraic sense which generalizes the well-known Heisenberg uncertainty relation. Two examples, the classical probability structure of Kolmogorov and the Hilbert space quantum logic, are given as illustrations throughout the whole discussion.
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.
In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.