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Peter Mittelstaedt's contributions to quantum logic and to the foundational problems of quantum theory have significantly realized the most authentic spirit of the International Quantum Structures Association: an original research about hard technical problems, which are often 'entangled' with the emergence of important changes in our general world-conceptions. During a time where both the logical and the physical community often showed a skeptical attitude towards Birkhoff and von Neumann's quantum logic, Mittelstaedt brought into light the deeply innovating features of a quantum logical thinking that allows us to overcome some strong and unrealistic assumptions of classical logical arguments. Later on his intense research on the unsharp approach to quantum theory and to the measurement problem stimulated the increasing interest for unsharp forms of quantum logic, creating a fruitful interaction between the work of quantum logicians and of many-valued logicians. Mittelstaedt's general views about quantum logic and quantum theory seem to be inspired by a conjecture that is today more and more confirmed: there is something universal in the quantum theoretic formalism that goes beyond the limits of microphysics, giving rise to interesting applications to a number of different fields. [ABSTRACT FROM AUTHOR]

Is it possible to give a logical characterization of entanglement and of entanglement-measures in terms of the probabilistic behavior of some gates? This question admits different (positive or negative) answers in the case of different systems of gates and in the case of different classes of density operators. In the first part of this article we investigate possible relations between entanglement-measures and the probabilistic behavior of quantum computational conjunctions. [ABSTRACT FROM AUTHOR]

Quantum computational logics represent a suitable abstract environment for modelling semantic situations where vagueness and ambiguity play a relevant role. In this framework vague possible worlds can be described as a kind of abstract scenes, where individual concepts, relational concepts and events behave as quantum-like superpositions that ambiguously allude to a number of possibly antagonistic alternatives. The quantum semantics can be naturally applied to a formal analysis of musical compositions, where both musical ideas and extra-musical meanings are characterized by some essentially vague and ambiguous features. We investigate some examples. [ABSTRACT FROM AUTHOR]

We study some properties of directoids and their expansions by additional signature, including bounded involutive directoids and complemented directoids. Among other results, we provide a shorter proof of the direct decomposition theorem for bounded involutive directoids given in Chajda and Länger (Directoids. An algebraic approach to ordered sets. Heldermann Verlag, Lemgo ); we present a description of central elements of complemented directoids; we show that the variety of directoids, as well as its expansions mentioned above, all have the strong amalgamation property. [ABSTRACT FROM AUTHOR]

Shi and Aharonov have shown that the Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. The basic algebraic properties of this system have been studied in Dalla Chiara et al. (Foundations of Physics 39(6):559-572, ), where we have introduced the notion of Shi-Aharonov quantum computational structure. In this paper we propose an algebraic abstraction from the Hilbert-space quantum computational structures, by introducing the notion of Toffoli-Hadamard algebra. From an intuitive point of view, such abstract algebras represent a natural quantum generalization of both classical and fuzzy-like structures. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to solve a problem posed in Di Nola et al. (Multiple Val. Logic 8:715-750, ) whether every pseudo BL-algebra with two negations is good, i.e. whether the two negations commute. This property is intimately connected with possessing a state, which in turn is essential in quantum logical applications. We approach the solution by describing the structure of pseudo BL-algebras and pseudo hoops as important families of quantum structures. We show when a pseudo hoop can be embedded into the negative cone of the reals. We give an equational base characterizing representable pseudo hoops. We also describe some subvarieties: normal-valued, and varieties where each maximal filter is normal. We produce some noncommutative covers and extend the area where each algebra is good. Finally, we show that there are uncountably many subvarieties of pseudo BL-algebras having members that are not good. [ABSTRACT FROM AUTHOR]

In the present paper we continue the investigation of the lattice of subvarieties of the variety of $${\sqrt{\prime}}$$ quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we show that the variety generated by the standard disk algebra D is not finitely based, and we provide an infinite equational basis for the same variety. [ABSTRACT FROM AUTHOR]

In the Fock space semantics, meanings of sentences are identified with density operators of the (unsymmetrized) Fock space $${\cal F}$$ based on the Hilbert space ℂ2. Generally, the meaning of a sentence is smeared over different sectors of $${\cal F}$$ . The standard quantum computational semantics is a limit case of the Fock space semantics, where the meaning of any sentence α only “lives” in one sector of $${\cal F}$$ , which is determined by the logical complexity of α. We prove that the global Fock space semantics and the standard quantum computational semantics characterize the same logic. [ABSTRACT FROM AUTHOR]

Two different generalizations of Brouwer–Zadeh posets (BZ posets) are introduced. The former (called pre-BZ poset) arises from topological spaces, whose standard power set orthocomplemented complete atomic lattice can be enriched by another complementation associating with any subset the set theoretical complement of its topological closure. This complementation satisfies only some properties of the algebraic version of an intuitionistic negation, and can be considered as, a generalized form of a Brouwer negation. The latter (called degenerate BZ poset) arises from the so-called special effects on a Hilbert space. It is shown that the standard Brouwer negation for effect operators produces a degenerate BZ poset with respect to the order induced from the partial sum operation. [ABSTRACT FROM AUTHOR]

It is shown that every effect algebra can be represented as a pasting of a systemwhere each element is the range of an unsharp observable. To describe the rangeof an unsharp observable algebraically, the notion of a “para-Booleanquasi-effect algebra” is introduced. Some intrinsic compatibility conditions ensuringcommensurability of effects are studied. [ABSTRACT FROM AUTHOR]