Maria Luisa Dalla Chiara

LOGICS FROM QUANTUM COMPUTATION

The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister (a system of qubits) or, more generally, with a mixture of quregisters (called qumix). In this framework, any sentence α of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister (qumix) associated to the atomic subformulas of α into the quregister (qumix) associated to α. A variant of the quantum computational semantics is represented by the quantum holistic semantics, which permits us to represent entangled meanings. Physical models of quantum computational logics can be built by means of Mach–Zehnder interferometers.

Paraconsistent quantum logics

Paraconsistent quantum logics are weak forms of quantum logic, where the noncontradiction and the excluded-middle laws are violated. These logics find interesting applications in the operational approach to quantum mechanics. In this paper, we present an axiomatization, a Kripke-style, and an algebraic semantical characterization for two forms of paraconsistent quantum logic. Further developments are contained in Giuntini and Greulings paper in this issue.

Quantum Computational Logics: A Survey

Quantum computation has suggested new forms of quantum logic, called quantum computational logics ([2]). The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, a system of qubits, representing a possible pure state of a compound quantum system. The generalization to mixed states, which might be useful to analyse entanglement-phenomena, is due to Gudder ([7]). Quantum computational logics represent non standard examples of unsharp quantum logic, where the non-contradiction principle is violated, while conjunctions and disjunctions are strongly non-idempotent. In this framework, any sentence a of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister associated to the atomic subforrnulas of α into the quregister associated to α.

Unsharp quantum logics

The event-structure of a state-event system, containing unsharp elements, can be described either as aregular involutive bounded poset, or alternatively as anunsharp orthoalgebra (called alsodifference poset oreffect algebra). Such structures give rise to different forms ofunsharp quantum logics.The event-structure of a state-event system, containing unsharp elements, can be described either as aregular involutive bounded poset, or alternatively as anunsharp orthoalgebra (called alsodifference poset oreffect algebra). Such structures give rise to different forms ofunsharp quantum logics.

Compositional and holistic quantum computational semantics

In quantum computational logic meanings of sentences are identified with quantum information quantities: systems of qubits or, more generally, mixtures of systems of qubits. We consider two kinds of quantum computational semantics: (1) a compositional semantics, where the meaning of a compound sentence is determined by the meanings of its parts; (2) a holistic semantics, which makes essential use of the characteristic “holistic” features of the quantum-theoretic formalism. We prove that the compositional and the holistic semantics characterize the same logic.

Fuzzy intuitionistic quantum logics

Fuzzy intuitionistic quantum logics (called also Brouwer-Zadeh logics) represent to non standard version of quantum logic where the connective “not” is split into two different negation: a fuzzy-like negation that gives rise to a paraconsistent behavior and an intuitionistic-like negation. A completeness theorem for a particular form of Brouwer-Zadeh logic (BZL3) is proved. A phisical interpretation of these logics can be constructed in the framework of the unsharp approach to quantum theory.

Epistemic Quantum Computational Structures in a Hilbert-space Environment

Quantum computation and quantum computational logics are intrinsically connected with some puzzling epistemic problems. In the framework of a quantum computational approach to epistemic logic we investigate the following question: is it possible to interpret the basic epistemic operations (having information, knowing) as special kinds of Hilbert-space operations? We show that non-trivial knowledge operations cannot be represented by unitary operators. We introduce the notions of strong epistemic quantum computational structure and of epistemic quantum computational structure, where knowledge operations are identified with special examples of quantum operations. This represents the basic tool for developing an epistemic quantum computational semantics, where epistemic sentences (like “Alice knows that the spin-value in the x-direction is up”) are interpreted as quantum pieces of information that may be stored by quantum objects.

Identity Questions from Quantum Theory

To what extent does quantum mechanics give rise to violations of Leibniz’ theory of identity? This is a highly controversial problem. With different arguments, the question has been positively answered by Margenau, van Fraassen, Mittelstaedt, da Costa, Krause and other scholars. A negative answer, instead, has been defended by Aerts, Piron, Costantini, Garibaldi.

Some foundational problems in mathematics suggested by physics

It is largely recognized that physics has represented an important source of suggestions in the historical development of mathematics and, in more recent times, even of logic. At the same time, if we look to the foundational investigations about both sciences in our century, one cannot help but notice that these researches have very rarely inter acted. Further, a number of discussions about the foundations of mathematics seem to go along with a somewhat naive and old fashioned image of physics, whereas foundational investigations about physics often propose an oversimplified image of the world of mathe matics (stressing for instance the characterization of mathematics as a merely analytical science). I would like to discuss in what sense contemporary physics might provide some interesting arguments and theoretical results, which could have a bearing on the foundational studies about mathematics. Let me start with a traditional philosophical question. It is well known that many discussions of the golden period of the foundational studies in mathematics in our century are essentially founded on a somewhat schematic contraposition between a platonistic view according to which, roughly, mathematical objects are described) and a concep tualista view (according to which mathematical objects are invented or constructed). Now, such a contraposition is clearly based on an extrapolation from the theoretical constructs of empirical sciences. Namely, the starting point of the traditional platonist in mathematics seems to be founded on a very uncritical concept of physical object, to which the platonist intends to assimilate even the concept of mathema tical object. In other words (according to the classical platonistic view), the concept of mathematical object should share the same logical and gnoseological characters as the concept of physical object. But what are the relevant properties of the concept of physical object? It is well known that contemporary physics has submitted to a strong criticism the traditional concept of object, developed by classical macrophysics [10]. I would like to recall only four main reasons, which in my opinion confirm the thesis according to which the concept of

Holistic logical arguments in quantum computation

Abstract 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.