Volume 66, Issue 2

Quantum computational logics represent a logical abstraction from the circuit-theory in quantum computation. In these logics formulas are supposed to denote pieces of quantum information (qubits, quregisters or mixtures of quregisters), while logical connectives correspond to (quantum logical) gates that transform quantum information in a reversible way. The characteristic holistic features of the quantum theoretic formalism (which play an essential role in entanglement-phenomena) can be used in order to develop a holistic version of the quantum computational semantics. In contrast with the compositional character of most standard semantic approaches, meanings of formulas are here dealt with as global abstract objects that determine the contextual meanings of the formulas’ components (from the whole to the parts ). We present a survey of the most significant logical arguments that are valid or that are possibly violated in the framework of this semantics. Some logical features that may appear prima facie strange seem to reflect pretty well informal arguments that are currently used in our rational activity.

In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let ∗ stand for the Łukasiewicz t-norm on [0, 1]2. Let us consider the operation ⋄ on [0, 1] by setting x ⋄ y = ( x ⊥ ∗ y ⊥ ) ⊥ ∗ ( x ∗ y ) ⊥ , where x ⊥ = 1− x . Let us call the triple L = ([0, 1], ⋄, 1) the Łukasiewicz groupoid. Let B be a Boolean algebra. Denote by L ( B ) the set of all L -valued measures ( L -valued states). We show as a main result of this paper that the family L ( B ) consists precisely of the union of classical real states and Z 2 -valued states. With the help of this result we characterize the L -valued states on orthomodular posets. Since the orthomodular posets are often understood as “quantum logics” in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any L -valued state.

A well known fact is that there is a finite orthomodular lattice with an order determining set of states which is not order embeddable into the standard quantum logic, the lattice L ( 𝓗 ) of all closed subspaces of a separable complex Hilbert space. We show that a finite generalized effect algebra is order embeddable into the standard effect algebra E ( 𝓗 ) of effects of a separable complex Hilbert space iff it has an order determining set of generalized states iff it is order embeddable into the power of a finite MV-chain. As an application we obtain an algorithm, which is based on the simplex algorithm, deciding whether such an order embedding exists and, if the answer is positive, constructing it.

We continue the study of symmetric logics , i.e., collections of subsets generalizing Boolean algebras and closed under the symmetric difference. We contribute to several open questions. One of them is whether there is á non-Boolean symmetric logic such that all states on it are Δ-subadditive.

The aim of the present paper is to study extension of functions defined on a subfamily of a D -poset E , as well as ν -measurability of elements in E . We have also considered monotone extensions for a nested sequence of generalized quantum spaces.

We apply the notion of generalized MV -algebra ( GMV -algebra, for short) in the sense introduced and studied by Galatos and Tsinakis. Let M be an upper bounded GMV -algebra. Then M can be constructed by using a pair ( G 1 , L ), where G 1 is a lattice ordered group and L is a filter on the lattice G1−$G_1^ - $ satisfying certain conditions. In the present paper we deal with the relations between convex subalgebras of M and convex ℓ -subgroups of G 1 .

In this paper we investigate a class of MV-algebras built up by fixing a Boolean algebra, one of its maximal ideals and an ℓ -group. We use such class to characterize, in the variety generated by all perfect MV-algebras, those MV-algebras which have their perfect skeleton as a quotient, giving an axiomatization of such a class and a representation theorem.

Let τ be a nonempty similarity type of algebras. A set H of τ -algebras is called rigid with respect to embeddability, if whenever A, B ∈ H and φ : A → B is an embedding, then A = B and φ is the identity map. We prove that if τ is a nonempty similarity type and 𝖒 is a cardinal such that no inaccessible cardinal is smaller than or equal to m, then there exists a set H of τ -algebras such that H is rigid with respect to embeddability and | H | = 𝖒. This result strengthens a result proved by the second author in 1980.

In a new algebraic approach, conformal algebras and vertex algebras are extended to two-sorted structures, with an additional component encoding the logical properties of locality. Within these algebras, locality is expressed as an identity, without the need for existential quantifiers. Two-sorted conformal algebras form a variety of two-sorted algebras, an equationally-defined class, and free conformal algebras are given by standard universal algebraic constructions. The variety of two-sorted conformal algebras is equivalent to a Mal’tsev variety of single-sorted algebras. Motivated by a question of Griess, subalgebras of reducts of conformal algebras are shown to satisfy a set of quasi-identities. The class of two-sorted vertex algebras does not form a variety, so open problems concerning the nature of that class are posed.

We prove a Carathéodory type extension theorem for σ -additive exhaustive modular measures on σ -continuous pseudo-D-lattices.

In this paper we study analytically a parameter switching (PS) algorithm applied to a class of systems of ODE, depending on a single real parameter. The algorithm allows the numerical approximation of any solution of the underlying system by simple periodical switches of the control parameter. Near a general approach of the convergence of the PS algorithm, some dissipative properties are investigated and the dynamical behavior of solutions is investigated with the Lyapunov function method. A numerical example is presented.

We define an index of compatibility for a probabilistic theory (PT). Quantum mechanics with index 0 and classical probability theory with index 1 are at the two extremes. In this way, quantum mechanics is at least as incompatible as any PT. We consider a PT called a concrete quantum logic that may have compatibility index strictly between 0 and 1, but we have not been able to show this yet. Finally, we show that observables in a PT can be represented by positive, vector-valued measures.

The paper deals with inner product spaces generated by states on Jordan algebras. We show an interplay between completeness of the Gelfand-Neumark-Segal representation space, geometric properties of states on Jordan algebras, structure of irreducible Jordan representations, and properties of normal states on second duals of Jordan algebras. We prove that if the GNS representation space is complete, then given state must be a convex combination of pure states. On the other hand, we analyze structure of inner product spaces arising from states on spin factors and Type I n , n ≥ 4, factors, showing their completeness as a consequence.

A synaptic algebra is a generalization of the self-adjoint part of a von Neumann algebra. For a synaptic algebra we study two weakened versions of commutativity, namely quasi-commutativity and operator commutativity, and we give natural conditions on the synaptic algebra so that each of these conditions is equivalent to commutativity. We also investigate the structure of a commutative synaptic algebra, prove that a synaptic algebra is commutative if and only if it is a vector lattice, and provide a functional representation for a commutative synaptic algebra.

A new type of mixtures of discrete probability distributions is presented. A family of discrete averaged mixed distributions is introduced. Its subclass of averaged mixed logarithmic distributions is analyzed. Probabilistic characterizations and connections with other types of mixing are derived. We show also some examples of the analyzed distributions found in literature.

In [HARDING, J.: Decompositions in quantum logic , Trans. Amer. Math. Soc. 348 (1996), 1839–1862] the author showed that the direct product decompositions of many different types of structures, such as sets, groups, vector spaces, topological spaces, and relational structures, naturally form orthomodular posets. When applied to the direct product decompositions of a Hilbert space, this construction yields the familiar orthomodular lattice of closed subspaces of the Hilbert space. In this note we consider orthomodular posets Fact X of decompositions of a finite set X . We consider the structure of these orthomodular posets, such as their size, shape, connectedness, states, and begin a study of their automorphism groups in the context of the natural map Γ from the group of permutations of X to the automorphism group of F ACT X . We show Γ is an embedding except when | X | is prime or 4, and completely describe the situation when | X | has two or fewer prime factors, when | X | = 2 3 and when | X | = 3 3 . The bulk of our effort lies in a series of combinatorial arguments to show Γ is an isomorphism when | X | = 27. We conjecture that this is the case whenever | X | has sufficiently many prime factors of sufficient size, and hope that our arguments here might be adapted to the general case.