Roberto Giuntini

The logical anomalies of quantum objects can be investigated in the semantics of quantum computational logics. One is dealing with new forms of quantum logic that have been inspired by quantum computation theories, where entanglement-phenomena play an important role. In this article we propose a semantic characterization for a weak form of first-order quantum computational logic with identity. We show how some intriguing questions that concern indiscernible quantum objects can be usefully analyzed in this logical framework.

Since its birth quantum mechanics has inspired new logical ideas. The most important “logical revolution” of the theory concerns a divergence between the concepts of maximal information and logically complete information. This is a natural consequence of Heisenberg’s uncertainty principle: any quantum pure state represents a maximal information that cannot be consistently extended to a richer knowledge about the physical system under consideration. At the same time, such information is logically incomplete, since it cannot decide all relevant properties of the system in question. Birkhoff and von Neumann’s quantum logic represents a natural logical abstraction from the algebraic structure of all events (that may occur to a quantum system $$\textbf{S}$$ ), mathematically represented as projection operators of the Hilbert space $$\varvec{\mathcal {H}}^{\textbf{S}}$$ associated to $$\textbf{S}$$. Projections of $$\varvec{\mathcal {H}}^{\textbf{S}}$$ represent sharp quantum events that satisfy the logical non-contradiction principle. Following Ludwig, the sharp concept of quantum event can be generalized to the unsharp concept of quantum effect, that gives rise to possible violations of the non-contradiction principle. The concepts of observable and of compatibility between observables have different behaviors in the framework of sharp and of unsharp quantum theory. Canonical examples of observables (say, position and momentum), that are incompatible in sharp quantum theory, turn out to be jointly measurable in the case of unsharp quantum theory. Interestingly enough, a famous thought experiment (proposed by Heisenberg) about the possibility of an approximate simultaneous measurement of position and momentum can be described and justified in the formal language of unsharp quantum theory.

How are abstract concepts formed and recognized on the basis of a previous experience? It is interesting to compare the behavior of human minds and of artificial intelligences with respect to this problem. Generally, a human mind that abstracts a concept (say, table), from a given set of known examples creates a table-Gestalt: a kind of vague and out of focus image that does not fully correspond to a particular table with well determined features. Can the construction of a gestaltic pattern, which is so natural for human minds, be taught to an intelligent machine? This problem can be successfully discussed in the framework of a quantum approach to machine learning. The basic idea is replacing classical datasets with quantum datasets where objects are described by special quantum states, involving the characteristic uncertainty and ambiguity of the quantum theoretic formalism. In this framework, the intuitive concept of Gestalt can be simulated by the mathematical concept of positive centroid of a given quantum dataset. Although recognition procedures are different for human and for artificial intelligences, there is a common method of “facing the problems” that seems to work in both cases. From a logical point if view, a quantum approach to machine learning can be reconstructed in the framework of a special form of quantum-logical semantics that has been suggested by quantum information theory.

The intrinsic informational content that characterizes quantum computational logics has naturally inspired some interesting and intriguing epistemic problems. We investigate the possibility of a quantum computational semantics for a first-order language that can express sentences like “Alice knows that everybody knows that she is pretty”. The basic question is: to what extent is it possible to interpret quantiers and epistemic operators as special examples of Hilbert-space operations? These logical operators turn out to have a similar logical behavior, giving rise to a “reversibility-breaking”. Unlike logical connectives, quantiers and epistemic operators can be represented as special operations that are generally irreversible (as happens in the case of measurement-procedures). An interesting feature of the epistemic quantum semantics is the failure of the unrealistic phenomenon of logical omniscience: Alice might know a given sentence without knowing all its logical consequences.

The theory of quantum logical circuits has naturally inspired new forms of quantum logic that have been termed quantum computational logics. From a semantic point of view, any formula of the language of a quantum computational logic is supposed to denote a piece of quantum information that lives in a Hilbert space whose dimension depends on the linguistic complexity of the formula in question. At the same time, the logical connectives are interpreted as special examples of quantum logical gates. Accordingly, any formula of a quantum computational language can be regarded as a synthetic logical description of a quantum circuit. In this way, linguistic formulas acquire a characteristic dynamic meaning, representing possible computational actions. The most natural semantics for quantum computational logics is a form of holistic semantics, where the puzzling entanglement-phenomena can be used as a logical resource. The concept of logical consequence, defined in this semantics, characterizes a weak form of quantum logic, where many important logical arguments (which are valid either in classical logic or in Birkhoff and von Neumann’s quantum logic) are possibly violated.

In spite of their strongly non-classical features, quantum computational logics include a subtheory that behaves classically, both from the logical and from the computational point of view. The holistic quantum computational semantics has a special fragment where pieces of information are represented by classical bits and registers, while the basic Boolean functions are represented as reversible operations. One can generalize this approach, by assuming a many-valued classical basis for quantum computation. In this way, qubits are replaced by qudits: quantum superpositions living in a Hilbert space whose dimension may be greater than two. The question whether the “qudit-semantics” (which gives rise to some interesting physical implementations) and the “qubit-semantics” characterize the same logic is an open problem.

This book provides an interdisciplinary approach to one of the most fascinating and important open questions in science: What is quantum mechanics really talking about? In the last decades quantum mechanics has given rise to a new quantum technological era, a revolution taking place today especially within the field of quantum information processing; which goes from quantum teleportation and cryptography to quantum computation. Quantum theory is probably our best confirmed physical theory. However, in spite of its great empirical effectiveness it stands today still without a universally accepted physical representation that allows us to understand its relation to the world and reality. The novelty of the book comes from the multiple perspectives put forward by top researchers in quantum mechanics, from Europe as well as North and South America, discussing the meaning and structure of the theory of quanta. The book comprises in a balanced manner physical, philosophical, logical and mathematical approaches to quantum mechanics and quantum information. Going from quantum superpositions and entanglement to dynamics and the problem of identity; from quantum logic, computation and quasi-set theory to the category approach and teleportation; from realism and empiricism to operationalism and instrumentalism; the book considers from different angles some of the most intriguing questions in the field. From Buenos Aires to Brussels and Cagliari, from Florence to Florianópolis, the interaction between different groups is reflected in the many different articles. This book is interesting not only to the specialists but also to the general public attempting to get a grasp on some of the most fundamental questions of present quantum physics.

How are abstract concepts and musical themes recognized on the basis of some previous experience? It is interesting to compare the different behaviors of human and of artificial intelligences with respect to this problem. Generally, a human mind that abstracts a concept (say, table) from a given set of known examples creates a table-Gestalt: a kind of vague and out of focus image that does not fully correspond to a particular table with well determined features. A similar situation arises in the case of musical themes. Can the construction of a gestaltic pattern, which is so natural for human minds, be taught to an intelligent machine? This problem can be successfully discussed in the framework of a quantum approach to pattern recognition and to machine learning. The basic idea is replacing classical data sets with quantum data sets, where either objects or musical themes can be formally represented as pieces of quantum information, involving the uncertainties and the ambiguities that characterize the quantum world. In this framework, the intuitive concept of Gestalt can be simulated by the mathematical concept of positive centroid of a given quantum data set. Accordingly, the crucial problem “how can we classify a new object or a new musical theme (we have listened to) on the basis of a previous experience?” can be dealt with in terms of some special quantum similarity-relations. Although recognition procedures are different for human and for artificial intelligences, there is a common method of “facing the problems” that seems to work in both cases.

The debate over the question whether quantum mechanics should be considered as a complete account of microphenomena has a long and deeply involved history, a turning point in which has been certainly the Einstein-Bohr debate, with the ensuing charge of incompleteness raised by the Einstein-Podolsky-Rosen argument. In quantum mechanics, physical systems can be prepared in pure states that nevertheless have in general positive dispersion for most physical quantities; hence in the EPR argument, the attention is focused on the question whether the account of the microphysical phenomena provided by quantum mechanics is to be regarded as an exhaustive description of the physical reality to which those phenomena are supposed to refer, a question to which Einstein himself answered in the negative. However, there is a mathematical side of the completeness issue in quantum mechanics, namely the question whether the kind of states with positive dispersion can be represented as a different, dispersion-free kind of states in a way consistent with the mathematical constraints of the quantum mechanical formalism. From this point of view, the other source of the completeness issue in quantum mechanics is the no hidden variables theorem formulated by John von Neumann in his celebrated book on the mathematical foundations of quantum mechanics, the preface of which already anticipates the program and the conclusion concerning the possibility of ‘neutralizing’ the statistical character of quantum mechanics

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.

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 some properties of two varieties of algebras arising from quantum computation - quasi-MV algebras and $\sqrt{^{\prime }}$ quasi-MV algebras - first introduced in \cite{Ledda et al. 2006}, \cite{Giuntini et al. 200+} and tightly connected with fuzzy logic. We establish the finite model property and the congruence extension property for both varieties; we characterize the quasi-MV reducts and subreducts of $\sqrt{^{\prime }}$ quasi-MV algebras; we give a representation of semisimple $\sqrt{^{\prime }}$ quasi-MV algebras in terms of algebras of functions; finally, we describe the structure of free algebras with one generator in both varieties.

We continue our investigation of paraorthomodular BZ*-lattices PBZ*-lattices, started in Giuntini et al., Mureşan. We shed further light on the structure of the subvariety lattice of the variety PBZL∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {PBZL}^{\mathbb {*}}$$\end{document} of PBZ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{*}$$\end{document}–lattices; in particular, we provide axiomatic bases for some of its members. Further, we show that some distributive subvarieties of PBZL∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {PBZL}^{\mathbb {*}}$$\end{document} are term-equivalent to well-known varieties of expanded KleeneKleene, C. lattices or of nonclassical modal algebrasNonclassical modal algebras. By so doing, we somehow help the reader to locate PBZ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{*}$$\end{document}–lattices on the atlas of algebraic structures for nonclassical logics.

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.

The general idea that inspires all approaches to quantum computation is that information can be stored and transmitted by quantum physical systems. Thus, the quantum-theoretic formalism represents the natural mathematical environment for quantum computation theory. While classical information theory (as well classical mechanics) are naturally based on a twovalued semantics, the characteristic uncertainties of the quantum world have brought about some deep logical innovations, due to the divergence between the concepts of maximal information and logically complete information. Quantum pure states represent pieces of information that are at the same time maximal (since they cannot be consistently extended to a richer knowledge) and logically incomplete (since they cannot decide all the relevant properties of the objects under investigation). This chapter illustrates the basic quantum-theoretic concepts that play an important role in quantum information and in quantum computation.