Can many-valued logic help to comprehend quantum phenomena?

Following {\L}ukasiewicz, we argue that future non-certain events should be described with the use of many-valued, not 2-valued logic. The Greenberger-Horne-Zeilinger `paradox' is shown to be an artifact caused by unjustified use of 2-valued logic while considering results of future non-certain events. Description of properties of quantum objects before they are measured should be performed with the use of propositional functions that form a particular model of infinitely-valued {\L}ukasiewicz logic. This model is distinguished by specific operations of negation, conjunction, and disjunction that are used in it.Comment: 10 pages, no figure

Quantum teleportation and Grover's algorithm without the wavefunction

In the same way as the quantum no-cloning theorem and quantum key distribution in two preceding papers, entanglement-assisted quantum teleportation and Grover's search algorithm are generalized by transferring them to an abstract setting, including usual quantum mechanics as a special case. This again shows that a much more general and abstract access to these quantum mechanical features is possible than commonly thought. A non-classical extension of conditional probability and, particularly, a very special type of state-independent conditional probability are used instead of Hilbert spaces and wavefunctions.Comment: 21 pages, including annex, important typo in annex corrected in v2, Found Phys (2017

A note on many valued quantum computational logics

The standard theory of quantum computation relies on the idea that the basic information quantity is represented by a superposition of elements of the canonical basis and the notion of probability naturally follows from the Born rule. In this work we consider three valued quantum computational logics. More specifically, we will focus on the Hilbert space C^3, we discuss extensions of several gates to this space and, using the notion of effect probability, we provide a characterization of its states.Comment: Pages 15, Soft Computing, 201

Weakly complete axiomatization of exogenous quantum propositional logic

A weakly complete finitary axiomatization for EQPL (exogenous quantum propositional logic) is presented. The proof is carried out using a non trivial extension of the Fagin-Halpern-Megiddo technique together with three Henkin style completions.Comment: 28 page

Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus

The Distributional Compositional Categorical (DisCoCat) model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambek's pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambek's previous type-logic, the Lambek calculus, which has been extensively used to formalise and reason about various linguistic phenomena. The apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings.Comment: 29 pages, pending publication in Annals of Pure and Applied Logi

Quantum key distribution without the wavefunction

A well-known feature of quantum mechanics is the secure exchange of secret bit strings which can then be used as keys to encrypt messages transmitted over any classical communication channel. It is demonstrated that this quantum key distribution allows a much more general and abstract access than commonly thought. The results include some generalizations for the Hilbert space version of quantum key distribution,but base upon a general non-classical extension of conditional probability. A special state-independent conditional probability is identified as origin of the superior security of quantum key distribution and may have more profound implications for the foundations and interpretation of quantum mechanics,quantum information theory, and the philosophical question what actually constitutes physical reality.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1502.0215

Theoretical Setting of Inner Reversible Quantum Measurements

We show that any unitary transformation performed on the quantum state of a closed quantum system, describes an inner, reversible, generalized quantum measurement. We also show that under some specific conditions it is possible to perform a unitary transformation on the state of the closed quantum system by means of a collection of generalized measurement operators. In particular, given a complete set of orthogonal projectors, it is possible to implement a reversible quantum measurement that preserves the probabilities. In this context, we introduce the concept of "Truth-Observable", which is the physical counterpart of an inner logical truth.Comment: 11 pages. More concise, shortened version for submission to journal. References adde

A presentation of Quantum Logic based on an "and then" connective

When a physicist performs a quantic measurement, new information about the system at hand is gathered. This paper studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the "and then" operation that combines old and new information. The "and then" connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the "and then" connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch (1969) satisfies all the properties studied in this paper. All properties satisfied by Finch's operation in modular lattices are valid in Hilbert Space Quantum Logic. It is not known whether all properties of Hilbert Space Quantum Logic are satisfied by Finch's operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean algebras are defined, ideals are characterized and a homomorphism theorem is proved.Comment: 28 pages. Submitte

Bell-type inequalities for bivariate maps on orthomodular lattices

Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (s-maps), are studied. It is shown that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our invesigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using s-maps and j-maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.Comment: 14 pages, no figure

An alternative Gospel of structure: order, composition, processes

We survey some basic mathematical structures, which arguably are more primitive than the structures taught at school. These structures are orders, with or without composition, and (symmetric) monoidal categories. We list several `real life' incarnations of each of these. This paper also serves as an introduction to these structures and their current and potentially future uses in linguistics, physics and knowledge representation.Comment: Introductory chapter to C. Heunen, M. Sadrzadeh, and E. Grefenstette. Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse. Oxford University Press, 201