M. Dalla Chiara

Individuals, Kinds and Names in Physics

It has been recognised for a long time that language plays a crucial role in science. The neopositivist philosophers were right when they put special emphasis on the analysis of the language of science. But is this analysis sufficient to clarify all the problems of the philosphy of science? Can science be reduced to a pure linguistic game — as Wittgenstein would put it — or is there something else? Surely science talks of something. Accordingly, the scientist must be aware of the relation existing between the language he uses and the things he is talking about. But here ends the ‘game’, if ever there was one, and formidable problems arise.

Partial and unsharp quantum logics

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.

Some Metalogical Pathologies of Quantum Logic

Quantum Logic (QL) gives rise to significant counterexamples to some metalogical properties which hold for classical logic (CL) and for a large class of weaker logics. A particularly interesting property is represented by the Lindenbaum property (LP), according to which any non-contradictory set of sentences can be extended to a non-contradictory and complete set, which for any sentence α has either α or the negation of α as its logical consequence. From an intuitive point of view, LP warrants a kind of weak semantical decidability, even in the case where the concept of truth of the logic under investigation does not satisfy the tertium-non datur principle. Metaphorically, one can always say “God knows, even if we do not know”; or -- if we want -- “God does not play dice”. On the contrary, the failure of L for a given class of logics, represents a kind of essential semantical undecidability.

Effect algebras and para-Boolean manifolds

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.

A many-valued approach to quantum computational logics

Quantum computational logics are special examples of quantum logic where formulas are supposed to denote pieces of quantum information (qubit-systems or mixtures of qubit-systems), while logical connectives are interpreted as reversible quantum logical gates. Hence, any formula of the quantum computational language represents a synthetic logical description of a quantum circuit. We investigate a many-valued approach to quantum information, where the basic notion of qubit has been replaced by the more general notion of qudit. The qudit-semantics allows us to represent as reversible gates some basic logical operations of Łukasiewicz many-valued logics. In the final part of the article we discuss some problems that concern possible implementations of gates by means of optical devices.

The Toffoli-Hadamard Gate System: an Algebraic Approach

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.

Quantum computational logic

Finally we study a new form of unsharp quantum logic that has been naturally suggested by the theory of quantum computation. One of the most interesting logical proposals that arise from quantum computation is the idea to use the quantum theoretical formalism in order to represent parallel reasoning.1 As is well known, the unit of measurement in classical information theory is the bit: one bit measures the information quantity that can be either transmitted or received whenever one chooses one element from a set consisting of two elements, say, from the set {0, 1}. From the intuitive point of view, both the objects 0 and 1 can be imagined as a well determined state of a classical physical system, for instance, the state of a tape cell in a given machine.

A fuzzy dynamic semantics for quantum histories

Time evolution in quantum theory (QT) gives rise to an intriguing situation. The greatest difficulty traces back to a famous conflict between two basic postulates of the theory: Schrödinger’s equation and von Neumann’s collapse of the wave function. After more than sixty years such a conflict seems to be far from being solved in a satisfactory way. As a consequence one can say that the concept of history of a quantum object seems to be a fairly ambiguous and fuzzy notion. Suppose a particle p (for instance an electron) evolving in a given time interval [t 0 , t 1 ]. Suppose the observer has associated to our particle, at time t 0 , a state s(t 0 ) that sums up his information about the system. A maximal information, that cannot be extended to a more precise one, corresponds to a pure state; otherwise we will have a mixed state. Now the Schrödinger equation determines a unique linear history Ss(t 0 ), 2 , i 2 , s(t1)T for our p, where each i represents the observer’s information about p at time t i . Whenever the initial state is pure, all the other states of the sequence will be pure. However, such a sequence does not represent the unique possible history for p. Even a pure state s(t i ) does not correspond to a logically complete knowledge: a number of relevant physical properties (represented by projections in the appropriate Hilbert space of the system) are not decided by i Let P be such an undecided property for i (for instance the ‘‘spin-value in the x-direction is up’’) and suppose the observer wants to check whether P in the ‘‘short’’ interval [t i , t j ]. Consequently, s(t i ) will be reduced to a new state u(t j ) by collapse of the wave function. Differently from s(t 0 ) and s(t i ), the state u(t j ) decides property P. This gives rise to a new history Su(t 0 ), 2 , i2 , u(t1)T, where any u(t i ) is determined by u(t j ) and by the Schrödinger equation. Such a plurality of possible histories seems to have ‘‘dangerous’’ consequences in the case of Einstein-PodolskyRosen like situations (briefly EPR) [4]. Let us refer to an abstract and simplified version of the EPR argument. We are dealing with two particles (say Sarah and Susan) that have interacted in the past (before time t 0 ) and are space-like separated since time t 0 (no signal can be sent from Sarah to Susan in the interval [t 0 , t 2 ] ). Suppose two observers: Oswald and Oscar. Oswald follows Sarah, while Oscar follows Susan. We are interested in two non compatible quantities P and Q that cannot be simultaneously measured; both P and Q can assume only two possible values: positive (]) and negative ([). Hence, we obtain two pairs of possible properties; say: MP`, P~N, MQ`, Q~N. Let P0 range over MP`, P~N. Similarly Q0. As a result of the past interaction, there is a strong correlation concerning the behaviour of Sarah and Susan with respect to the properties P 0 and Q0 :

Back to Hilbert space

We will now return to orthodox QT. We will compare the abstract event-state systems, studied in the previous chapter, with the concrete examples that emerge in the framework of Hilbert space structures.

Abstract axiomatic foundations of sharp QT

We will now develop an abstract analysis of the basic conceptual structures of QT in a framework that is relatively independent of the Hilbert space formalism. On this basis, orthodox QT will be reconstructed as a particular model of such a general formal approach.