Roberto Giuntini

The quantum computational semantics can be naturally applied to fields (far apart from microphysics) where ambiguity, vagueness, holism, and contextuality play an essential role. On this basis one can develop a general theory of vague possible worlds, which allows us to understand some abstract reasons why a “metaphorical thinking” is often successfully used in the languages of art and sometimes even in the field of exact sciences. A significant case is represented by musical languages, whose semantics can be formally analyzed by using some basic concepts of the quantum-theoretic formalism. In this framework, both musical thoughts and extra-musical meanings (evoked by musical compositions) can be represented as special examples of quantum superpositions: ambiguous ideas that essentially allude to a “cloud” of alternative possibilities.

After Feynman’s pioneering work, the abstract mathematical model for quantum computers has been often represented in terms of the notion of quantum Turing machine (the quantum counterpart of the classical notion of Turing machine). But what exactly are quantum Turing machines? So far, the literature has not provided a rigorous “institutional” concept of quantum Turing machine. Some definitions seem to be based on a kind of “imitation” of the classical definition of Turing machine, by referring to a tape (where the symbols are written) and to a moving head (which changes its position on the tape). These concepts, however, seem to be hardly applicable to physical quantum computers, where quantum uncertainties should also concern the behavior of moving heads. We introduce the more abstract notion of quantum computing machine, whose computations can be linguistically described by formulas of quantum computational logics. To what extent can abstract quantum computing machines be simulated by classical Turing machines? Does quantum computation give rise to possible violations of the Church-Turing thesis? These hard questions did not find, so far, a definite answer.

Entanglement represents one of the basic features of quantum theory, which has for a long time been described as “mysterious” and “potentially paradoxical”. Although the term “entanglement” was not explicitly used by Einstein, Podolsky and Rosen in their celebrated article “Can quantum mechanical description of reality be considered complete?”, entangled pure states are essentially involved in the paradoxical situation discussed in their paper. Chapter 3 is devoted to a logical analysis of the Einstein-Podolsky-Rosen paradox (in jargon “EPR-paradox”). It is shown how “EPR-situations” have later on been transformed into powerful resources, even from a technological point of view. As a significant example, teleportation-experiments are briefly illustrated.

Researches in the field of quantum information and quantum computation have naturally stimulated new debates about foundational and philosophical problems of quantum theory. “Information interpretations” according to which quantum theory should be mainly regarded as a “revolutionary information theory” have been opposed to more traditional “realistic” assumptions, according to which the pure states of the quantumtheoretic formalism shall always “mirror” objective properties of physical systems that exist (or may exist) in the real world. We analyze some different point of views that have been proposed and discussed in these debates.

The basic idea of quantum computer theory is that computations can be performed by some quantum systems that evolve in time. Accordingly, by applying Schrodinger’s equation, it is natural to assume that quantum information is processed by special examples of unitary operators (called quantum logical gates), which transform in a reversible way the pure states of the quantum systems that store the information in question. This chapter gives a synthetic presentation of the main “mathematical characters” of the quantum computational game: qubits, quregisters, mixtures of quregisters, quantum logical gates. The last Section of the Chapter illustrates possible physical implementations of some quantum logical circuits by means of special variants of the Mach-Zehnder interferometer, an apparatus that has played an important role in the philosophical debates about quantum theory.

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.

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 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.

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.

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.

Constructibility and complexity play central roles in recent research in computer science, mathematics and physics. For example, scientists are investigating the complexity of computer programs, constructive proofs in mathematics and the randomness of physical processes. But there are different approaches to the explication of these concepts. This volume presents important research on the state of this discussion, especially as it refers to quantum mechanics. This `foundational debate' in computer science, mathematics and physics was already fully developed in 1930 in the Vienna Circle. A special section is devoted to its real founder Hans Hahn, referring to his contribution to the history and philosophy of science. The documentation section presents articles on the early Philipp Frank and on the Vienna Circle in exile. Reviews cover important recent literature on logical empiricism and related topics.

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.

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.

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.