Logics from ' quasi-MV algebras (original) (raw)

MV-Algebras and Quantum Computation

Studia Logica, 2006

We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a " ‡at" quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers.

Expanding quasi-MV algebras by a quantum operator

2007

We investigate an expansion of quasi-MV algebras ([10]) by a genuine quantum unary operator. The variety √ QMV of such √ quasi-MV algebras has a subquasivariety whose members-called cartesian-can be obtained in an appropriate way out of MV algebras. After showing that cartesian √ quasi-MV algebras generate √ QMV, we prove a standard completeness theorem for √ QMV w.r.t. an algebra over the complex numbers.

The Logic of Quasi-MV Algebras

Journal of Logic and Computation, 2010

The algebraic theory of quasi-MV algebras, generalisations of MV algebras arising in quantum computation, is by now rather well-developed. Although it is possible to define several interesting logics from these structures, so far this aspect has not been investigated. The present paper aims at filling this gap.

Algebraic Methods in Quantum Logic

This monograph gives an overview of main results obtained within the solution of the project "Algebraic methods in quantum logic", being realized in the cooperation of the Faculty of Science of Masaryk University in Brno and the Faculty of Science of Palacký University in Olomouc. An innovated approach of the studied branch of mathematics consists of several chapters. The first chapter comprises of summarization of the results. The further chapters deal with more detailed and illustrated view of the studied topic.

Some generalizations of fuzzy structures in quantum computational logic

International Journal of General Systems, 2011

Quantum computational logics provide a fertile common ground for a uni ed treatment of vagueness and uncertainty. In this paper we describe an approach to the logic of quantum computation that has been recently taken up and developed by the present authors. Special attention will be devoted to a generalisation of Chang's MV algebras (called quasi-MV algebra) which abstracts over the algebra whose universe is the set of qumixes of the 2-dimensional complex Hilbert space, as well as to its expansions by additional quantum connectives. We furthermore explore some future research perspectives, also in the light of some recent limitative results whose general signi cance will be duly assessed. We thank Hector Freytes and Marisa Dalla Chiara for the stimulating conversations on the topics covered in this paper.

An Algebra of Quantum Processes

2010

We introduce an algebra qCCS of pure quantum processes in which no classical data is involved, communications by moving quantum states physically are allowed, and computations is modeled by super-operators. An operational semantics of qCCS is presented in terms of (non-probabilistic) labeled transition systems. Strong bisimulation between processes modeled in qCCS is defined, and its fundamental algebraic properties are established, including uniqueness of the solutions of recursive equations. To model sequential computation in qCCS, a reduction relation between processes is defined. By combining reduction relation and strong bisimulation we introduce the notion of strong reduction-bisimulation, which is a device for observing interaction of computation and communication in quantum systems. Finally, a notion of strong approximate bisimulation (equivalently, strong bisimulation distance) and its reduction counterpart are introduced. It is proved that both approximate bisimilarity and...

Operator Algebras and the Foundations of Quantum Mechanics

The purpose of this thesis is to analyse the Hilbert Space requirement for Quantum Mechanics. In particular, we justify sharp observables but question the requirement of completeness of the inner product space and the underlying Öeld. We view our mathematical framework as a dynamical theory but with a mysterious probabilistic interpretation instead of the otherway round. Whenever we speak of "Quantum Mechanics", we mean Non-relativistic Quantum Mechanics. To make things less messy, we assume associativity through-out. No attempt has been made to refer to QFT and statistical quantum mechanics and we use conventional mathematical symbols instead of Diracís formalism.

Quantum computational logic with mixed states

Mathematical Logic Quarterly, 2013

Using an algebraic framework we solve a problem posed in [5] and [7] about the axiomatizability of a type quantum computational logic related to fuzzy logic. A Hilbert-style calculus is developed obtaining an algebraic strong completeness theorem. 1 a qbit. A qbit state (the quantum counterpart of the classical bit) is represented by a unit vector in C 2 and, generalizing for a positive integer n, n-qbits are pure states represented by unit vectors in C 2 n . They conform the information units in quantum computation. These state spaces only concerned with the "static" part of quantum computing and possible logical systems can be founded in the Birkhoff and von Neumann quantum logic based on the Hilbert lattices L(C 2 n ) [11]. Similarly to the classical computing case, we can introduce and study the behavior of a number of quantum logical gates (hereafter quantum gates for short) operating on qbits, giving rise to "new forms" of quantum logic. These gates are mathematically represented by unitary operators on the appropriate Hilbert spaces of qbits. In other words, standard quantum computation is mathematically founded on "qbits-unitary operators" and only takes into account reversible processes. This framework can be generalized to a powerful mathematical representation of quantum computation in which the qbit states are replaced by density operators over Hilbert spaces and unitary operators by linear operators acting over endomorphisms of Hilbert spaces called quantum operations. The new model "density operators-quantum operations" also called "quantum computation with mixed states" ([1, 32]) is equivalent in computational power to the standard one but gives a place to irreversible processes as measurements in the middle of the computation.

Algebras of measurements: the logical structure of quantum mechanics

International Journal of …, 2006

In Quantum Physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties of such operators are justified on epistemological grounds. Commutation of measurements is a central topic of interest. Classical logical systems may be viewed as measurement algebras in which all measurements commute.