Roberto Giuntini

The characteristic holistic features of the quantum theoretic formalism and the intriguing notion of entanglement can be applied to a field that is far from microphysics: logical semantics. Quantum computational logics are new forms of quantum logic that have been suggested by the theory of quantum logical gates in quantum computation. In the standard semantics of these logics, sentences denote quantum information quantities: systems of qubits (quregisters) or, more generally, mixtures of quregisters (qumixes), while logical connectives are interpreted as special quantum logical gates (which have a characteristic reversible and dynamic behavior). In this framework, states of knowledge may be entangled, in such a way that our information about the whole determines our information about the parts; and the procedure cannot be, generally, inverted. In spite of its appealing properties, the standard version of the quantum computational semantics is strongly “Hilbert-space dependent”. This certainly represents a shortcoming for all applications, where real and complex numbers do not generally play any significant role (as happens, for instance, in the case of natural and of artistic languages). We propose an abstract version of quantum computational semantics, where abstract qumixes, quregisters and registers are identified with some special objects (not necessarily living in a Hilbert space), while gates are reversible functions that transform qumixes into qumixes. In this framework, one can give an abstract definition of the notions of superposition and of entangled pieces of information, quite independently of any numerical values. We investigate three different forms of abstract holistic quantum computational logic

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, 2009), 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

Shi and Aharonov have shown that the Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. We study the basic algebraic properties of this system by introducing the notion of Shi-Aharonov quantum computational structure. We show that the quotient of this structure is isomorphic to a structure based on a particular set of complex numbers (the closed disc with center \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\frac{1}{2},\frac{1}{2})$\end{document} and radius \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{1}{2}$\end{document} ).

We investigate certain Brouwer-Zadeh lattices that serve as abstract counterparts of lattices of effects in Hilbert spaces under the spectral ordering. These algebras, called PBZ*-lattices, can also be seen as generalisations of orthomodular lattices and are remarkable for the collapse of three notions of “sharpness” that are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ*-lattices and their reducts; in particular, we prove some embedding results for PBZ*-lattices and provide an initial description of the lattice of PBZ*-varieties.

The total and the sharp character of orthodox quantum logic has been put in question in different contexts. This paper presents the basic ideas for a unified approach to partial and unsharp forms of quantum logic. We prove a completeness theorem for some partial logics based on orthoalgebras and orthomodular posets. We introduce the notion of unsharp orthoalgebra and of generalized MV algebra. The class of all effects of any Hilbert space gives rise to particular examples of these structures. Finally, we investigate the relationship between unsharp orthoalgebras, generalized MV algebras, and orthomodular lattices

This paper will take into account the Lindenbaum property in Orthomodular Quantum Logic (OQL) and Partial Classical Logic (PCL). The Lindenbaum property has an interest both from a logical and a physical point of view since it has to do with the problem of the completeness of quantum theory and with the possibility of extending any semantically non-contradictory set of formulas to a semantically non-contradictory complete set of formulas. The main purpose of this paper is to show that both OQL and PCL cannot satisfy the Lindenbaum property.

Some algebraic structures determined by the class σ(þ) of all effects of a Hilbert space þ and by some subclasses of σ(þ) are investigated, in particular de Morgan-Brouwer-Zadeh posets [it is proved that σ(þ n )(n<∞) has such a structure], Brouwer-Zadeh * posets (a quite trivial example consisting of suitable effects is given), and Brouwer-Zadeh 3 posets which are both de Morgan and *.It is shown that a nontrivial class of effects of a Hilbert space exists which is a BZ 3 poset. An ɛ-preclusivity relation on the set of all vectors of þ is introduced, and it is shown that it satisfies the regularity condition also for ε∃ [1/2, 1]

This book provides a general survey of the main concepts, questions and results that have been developed in the recent interactions between quantum information, quantum computation and logic. Divided into 10 chapters, the books starts with an introduction of the main concepts of the quantum-theoretic formalism used in quantum information. It then gives a synthetic presentation of the main “mathematical characters” of the quantum computational game: qubits, quregisters, mixtures of quregisters, quantum logical gates. Next, the book investigates the puzzling entanglement-phenomena and logically analyses the Einstein–Podolsky–Rosen paradox and introduces the reader to quantum computational logics, and new forms of quantum logic. The middle chapters investigate the possibility of a quantum computational semantics for a language that can express sentences like “Alice knows that everybody knows that she is pretty”, explore the mathematical concept of quantum Turing machine, and illustrate some characteristic examples that arise in the framework of musical languages. The book concludes with an analysis of recent discussions, and contains a Mathematical Appendix which is a survey of the definitions of all main mathematical concepts used in the book.

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

In 1920 Łukasiewicz published a two-page article whose title was “On Three-valued Logic”. The paper proposes a semantic characterization for the logic that has been later called Ł3 . In spite of the shortness of the paper, all the important points concerning the semantics of Ł3 are already there and can be naturally generalized to the case of a generic number n of truth-values . The conclusion of the article is quite interesting:The present author is of the opinion that three-valued logic has above all theoretical importance as an endeavour to construct a system of non-aristotelian logic. Whether the new system of logic has any practical importance will be seen only when the logical phenomena, especially those in the deductive sciences, are thoroughly examined, and when the consequences of the indeterministic philosophy, which is the metaphysical substratum of the new logic, can be compared with empirical data

The debate about constructivism in physics has led to different kinds of questions that can be conventionally framed in two classes. One concerns the mathematics that is considered for the theoretical development of physics. The other is concerned with the experimental parts of physical theories. It is unnecessary to observe that the intersection between our two classes of problems is far from being empty. In this paper we will mainly deal with topics belonging to the second class. However, let us briefly mention some important problems that have been debated in the framework of our first class. For instance, the following: to what extent do the undecidability and incompleteness results of classical mathematics affect fragments of physical theories, in such way as to have a “real physical meaning”? are the mathematical arguments that seem to be essential for physics justifiable in the framework of traditional mathematical constructivism?The first question has recently been investigated by Pitowski, Penrose, da Costa, Doria, Mundici, Svozil and others. As expected by most logicians, one can construct undecidable sentences whose physical meaning seems to be hardly questionable. This happens both in classical and in quantum mechanics

In the standard approach to quantum mechanics, closed subspaces of a Hilbert space represent propositions. In the operational approach, closed subspaces are replaced by effects that represent a mathematical counterpart for properties which can be measured in a physical system. Effects are a proper generalization of closed subspaces. Effects determine a Brouwer-Zadeh poset which is not a lattice. However, such a poset can be embedded in a complete Brouwer-Zadeh lattice. From an intuitive point of view, one can say that these structures represent a natural logical abstraction from the structure of propositions of a quantum system. The logic that arises in this way is Brouwer-Zadeh logic. This paper shows that such a logic can be characterized by means of an algebraic and a Kripkean semantics. Finally, a strong completeness theorem for BZL is proved

The main aim of the paper is to solve a problem posed in Di Nola et al. (Multiple Val. Logic 8:715–750, 2002) 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