Sylvia Pulmannová

New trends in quantum structures

Preface. Introduction. 1. D-posets and Effect Algebras. 2. MV-algebras and QMV-algebras. 3. Quotients of Partial Abelian Monoids. 4. Tensor Product of D-Posets and Effect Algebras. 5. BCK-algebras. 6. BCK-algebras in Applications. 7. Loomis-Sikorski Theorems for MV-algebras and BCK-algebras. Bibliography. Index of Symbols. Index. List of Figures and Tables.

The center of an effect algebra

An effect algebra is a partial algebra modeled on the standard effect algebra of positive self-adjoint operators dominated by the identity on a Hilbert space. Every effect algebra is partially ordered in a natural way, as suggested by the partial order on the standard effect algebra. An effect algebra is said to be distributive if, as a poset, it forms a distributive lattice. We define and study the center of an effect algebra, relate it to cartesian-product factorizations, determine the center of the standard effect algebra, and characterize all finite distributive effect algebras as products of chains and diamonds.

Difference posets, effects, and quantum measurements

Difference posets as generalizations of quantum logics, orthoalgebras, and effects are studied. Observables and measures generalizing normalized POV-measures and generalized measures on sets of effects are introduced. Characterization of orthomodularity of subsets of a difference poset in terms of triangle closedness and regularity of these subsets enables us to characterize observables with a Boolean range. Boolean powers of difference posets are investigated; they have similar properties to that of tensor products, and their connection with quantum measurements is studied.

Orthocomplete effect algebras

We prove that for every orthocomplete effect algebra E the center of E forms a complete Boolean algebra. As a consequence, every orthocomplete atomic effect algebra is a direct product of irreducible ones.

Tensor product of quantum logics

A quantum logic is the couple (L,M) where L is an orthomodular σ‐lattice and M is a strong set of states on L. The Jauch–Piron property in the σ‐form is also supposed for any state of M. A ‘‘tensor product’’ of quantum logics is defined. This definition is compared with the definition of a free orthodistributive product of orthomodular σ‐lattices. The existence and uniqueness of the tensor product in special cases of Hilbert space quantum logics and one quantum and one classical logic are studied.

Convex and linear effect algebras

Abstract It is shown that convex effect algebras arise naturally in the description of a physical statistical system. An effect algebra that is a convex subset of a real linear space is called a linear effect algebra and it is demonstrated that any convex effect algebra is affinely isomorphic to a linear effect algebra. Convex effect algebras that possess separating and order determining state spaces are characterized. It is shown that an effect algebra P is imbeddable in an interval of an order unit space if and only if the state space of P is order determining. Sharp and extreme elements of convex effect algebras are studied and compared. An alternative definition of a convex effect algebra called a CE-algebra is considered and MV-algebras are discussed.

Bell inequalities on quantum logics

Three types of Bell inequalities, expressed in terms of quantum logics (orthomodular lattices) and states on them, are studied. It is shown that the two first types are equivalent to the subadditivity of the corresponding state. The third type is equivalent to the existence of a Boolean quotient B of the orthomodular lattice L and a state s on B, such that s⋅φ=s, where s is the state on L and φ:L→B is the quotient mapping.

D-test spaces and difference posets☆

Abstract D-test spaces, generalizing known test spaces, are introduced, and their connection to difference posets is presented. In addition, a tensor product of D-test spaces and their relationship to the tensor product of corresponding difference posets is studied.

EFFECT ALGEBRAS OF POSITIVE LINEAR OPERATORS DENSELY DEFINED ON HILBERT SPACES

We show that the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space can be equipped with partial sum of operators making it a generalized effect algebra. This sum coincides with the usual sum of two operators whenever it exists. Moreover, blocks of this generalized effect algebra are proper sub-generalized effect algebras. All intervals in this generalized effect algebra become effect algebras which are Archimedean, convex, interval effect algebras, for which the set of vector states is order determining. Further, these interval operator effect algebras possess faithful states.

Remarks on the order for quantum observables

Relations between generalized effect algebras and the sets of classical and quantum observables endowed with an ordering recently introduced in [GUDDER, S.: An order for quantum observables, Math. Slovaca 56 (2006), 573–589] are studied. In the classical case, a generalized OMP, while in the quantum case a weak generalized OMP is obtained. Existence of infima for arbitrary sets and suprema for above bounded sets in the quantum case is shown. Compatibility in the sense of Mackey is characterized.