An Unsharp Logic from Quantum Computation

Arxiv preprint quant-ph/0101028, 2001

We investigate some forms of quantum logic arising from the standard and the unsharp approach.

International Journal of Quantum Information, 2005

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.

Trends in Logic, 2003

Quantum computation has suggested new forms of quantum logic, called quantum computational logics ([CDCGL02]). 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 ([Gu02]). 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 α of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister associated to the atomic subformulas of α into the quregister associated to α.

Trends in logic, 2018

VOLUME 48 The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporary formal logic and its applications and relations to other disciplines. The series aims at publishing monographs and thematically coherent volumes dealing with important developments in logic and presenting significant contributions to logical research. The series is open to contributions devoted to topics ranging from algebraic logic, model theory, proof theory, philosophical logic, non-classical logic, and logic in computer science to mathematical linguistics and formal epistemology. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time.

Journal of Philosophical Logic

We begin by discussing the history of quantum logic, dividing it into three eras or lives. The first life has to do with Birkhoff and von Neumann's algebraic approach in the 1930's. The second life has to do with attempt to understand quantum logic as logic that began in the late 1950's and blossomed in the 1970's. And the third life has to do with recent developments in quantum logic coming from its connections to quantum computation. We discuss our own work connecting quantum logic to quantum computation (viewing quantum logic as the logic of quantum registers storing qubits), and make some speculations about mathematics based on quantum principles.

2006

Quantum theory is widely accepted as the most successful theory of natural science. One of the most important consequences of the birth of quantum mechanics (QM), especially in its Hilbert space formalism, has been the development of a new kind of logic, called quantum logic (QL). The logic underlying classical theory is a Boolean (two-valued) logic, while quantum logic proposed as a logic of quantum mechanics is currently understood as a logic whose truth values are taken from an orthomodular lattice. The major difference relies on distributivity.

Article CITATIONS 0 READS 17 2 authors, including: Some of the authors of this publication are also working on these related projects: interpretation of analytical mechanics through the two dichotomies. Search of a new formualtion of quantum mechanics relying on the alternative choices of the Dirac-von Neumann's one A new View project Antonino Drago University of Naples Federico II 70 PUBLICATIONS 88 CITATIONS SEE PROFILE All content following this page was uploaded by Antonino Drago on 14 January 2015.

The Journal of Symbolic Logic, 2005

Our understanding of Nature comes in layers, so should the development of logic. Classic logic is an indispensable part of our knowledge, and its interactions with computer science have recently dramatically changed our life. A new layer of logic has been developing ever since the discovery of quantum mechanics. G. D. Birkhoff and von Neumann introduced quantum logic in a seminal paper in 1936 [BV]. But the definition of quantum logic varies among authors (see [CG]). How to capture the logic structure inherent in quantum mechanics is very interesting and challenging. Given the close connection between classical logic and theoretical computer science as exemplified by the coincidence of computable functions through Turing machines, recursive function theory, and λ-calculus, we are interested in how to gain some insights about quantum logic from quantum computing. In this note we make some observations about quantum logic as motivated by quantum computing (see [NC]) and hope more people will explore this connection.

Electronic Proceedings in Theoretical Computer Science, 2012

In this paper, we consider Turing machines based on unsharp quantum logic. For a lattice-ordered quantum multiple-valued (MV) algebra E , we introduce E-valued non-deterministic Turing machines (E NTMs) and E-valued deterministic Turing machines (E DTMs). We discuss different Evalued recursively enumerable languages from width-first and depth-first recognition. We find that width-first recognition is equal to or less than depth-first recognition in general. The equivalence requires an underlying E value lattice to degenerate into an MV algebra. We also study variants of E NTMs. E NTMs with a classical initial state and E NTMs with a classical final state have the same power as E NTMs with quantum initial and final states. In particular, the latter can be simulated by E NTMs with classical transitions under a certain condition. Using these findings, we prove that E NTMs are not equivalent to E DTMs and that E NTMs are more powerful than E DTMs. This is a notable difference from the classical Turing machines.

2018

In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures. 1Universidad Nacional de La Plata, Instituto de F́ısica (IFLP-CCT-CONICET), C.C. 727, 1900 La Plata, Argentina 2Universitá di Cagliari, Via Is Mirrionis 1, 09123, Cagliari-Italia 3Center Leo Apostel for Interdisciplinary Studies and, Department of Mathematics, Brussels Free University Krijgskundestraat 33, 1160 Brussels, Belgium