Quantum logic as a dynamic logic

We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear "no". Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth's proposal of applying Tarski's semantical methods to the analysis of physical theories, with an empirical-experimental approach to Logic, as advocated by both Beth and Putnam, but understood by us in the view of the operational-realistic tradition of Jauch and Piron, i.e. as an investigation of "the logic of yes-no experiments" (or "questions"). Technically, we use the recently-developed setting of Quantum Dynamic Logic (Baltag and Smets 2005, 2008) to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results (Baltag and Smets 2005), we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the "non-classicality" of quantum behavior than any perspective based on static Propositional Logic

Ultra-Low-Power Superconductor Logic

We have developed a new superconducting digital technology, Reciprocal Quantum Logic, that uses AC power carried on a transmission line, which also serves as a clock. Using simple experiments we have demonstrated zero static power dissipation, thermally limited dynamic power dissipation, high clock stability, high operating margins and low BER. These features indicate that the technology is scalable to far more complex circuits at a significant level of integration. On the system level, Reciprocal Quantum Logic combines the high speed and low-power signal levels of Single-Flux- Quantum signals with the design methodology of CMOS, including low static power dissipation, low latency combinational logic, and efficient device count.Comment: 7 pages, 5 figure

Łukasiewicz-Moisil Many-Valued Logic Algebra of Highly-Complex Systems

A novel approach to self-organizing, highly-complex systems (HCS), such as living organisms and artificial intelligent systems (AIs), is presented which is relevant to Cognition, Medical Bioinformatics and Computational Neuroscience. Quantum Automata (QAs) were defined in our previous work as generalized, probabilistic automata with quantum state spaces (Baianu, 1971). Their next-state functions operate through transitions between quantum states defined by the quantum equations of motion in the Schroedinger representation, with both initial and boundary conditions in space-time. Such quantum automata operate with a quantum logic, or Q-logic, significantly different from either Boolean or Łukasiewicz many-valued logic. A new theorem is proposed which states that the category of quantum automata and automata--homomorphisms has both limits and colimits. Therefore, both categories of quantum automata and classical automata (sequential machines) are bicomplete. A second new theorem establishes that the standard automata category is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (M,R)--Systems which are open, dynamic biosystem networks with defined biological relations that represent physiological functions of primordial organisms, single cells and higher organisms

Logics of Informational Interactions

The pre-eminence of logical dynamics, over a static and purely propositional view of Logic, lies at the core of a new understanding of both formal epistemology and the logical foundations of quantum mechanics. Both areas appear at first sight to be based on purely static propositional formalisms, but in our view their fundamental operators are essentially dynamic in nature. Quantum logic can be best understood as the logic of physically-constrained informational interactions (in the form of measurements and entanglement) between subsystems of a global physical system. Similarly, (multi-agent) epistemic logic is the logic of socially-constrained informational interactions (in the form of direct observations, learning, various forms of communication and testimony) between “subsystems” of a social system. Dynamic Epistemic Logic (DEL) provides us with a unifying setting in which these informational interactions, coming from seemingly very different areas of research, can be fully compared and analyzed. The DEL formalism comes with a powerful set of tools that allows us to make the underlying dynamic/interactive mechanisms fully transparent

New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Intuitionistic logic, in which the double negation law not-not-P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold. The algebraic notions of effect algebra/module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X -> 1+1 carry such effect module structure, and can be used as predicates. Predicates are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C*-algebra or Hilbert space. In quantum foundations the duality between states and effects plays an important role. It appears here in the form of an adjunction, where we use maps 1 -> X as states. For such a state s and a predicate p, the validity probability s |= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature. Measurement from quantum mechanics is formalised categorically in terms of `instruments', using L\"uders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C*-algebras, side-effect appear exclusively in the non-commutative case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems

Combining paraconsistent and dynamic logic for Qiskit

Dissertação de mestrado integrado em Engenharia FísicaThis dissertation introduces a logic aimed at combining dynamic logic and paraconsistent logic for application to the quantum domain, to reason about quantum phase properties: Paraconsistent Phased Logic Of Quantum Programs (PhLQP◦ ). In the design PhLQP◦ , firstly the dynamic was built first, Phased Logic Of Quantum Programs (PhLQP). PhLQP is itself a dynamic logic capable of dealing with quantum phase properties, quantum measurements, unitary evolutions, and entanglements in compound systems , since it is a redesign of the already existing Logic Of Quantum Programs (LQP), [14], over a representation of quantum states restricted to a space B equipped with only two computational basis, standard and Hadamard. As instances of applications of the logic PhLQP, there is a formal proof of the correctness of the Quantum Teleportation Protocol, of the 2-party and 4-party of the Quantum Leader Election (QLE) protocol, and of the Quantum Fourier Transform (QFT) operator for 1, 2 and 3 qubits . On a second stage, PhLQP was extended with the connective ◦ known as the consistency operator, a typical connective of the paraconsistent logics Logics of Formal Inconsistency (LFIs), [8, 21, 22]. The definition of consistent quantum state and a set of proper para consistent axioms for the quantum domain, Fundamental Paraconsistent Quantum Axioms (FParQAxs), were provided. An example of application of PhLQP◦ is the possibility of express and prove correctness of the universal quantum gate, the Deustch gate.Esta dissertação introduz uma lógica que tem como objectivo combinar lógica dinâmica e lógica paraconsistente com aplicação no domínio quântico, assim como expressar propriedades relacionadas com fases quânticas: PhLQP◦. No projetar da PhLQP◦, primeiramente concebeu-se a sua componente dinâmica, PhLQP. PhLQP por si só é uma lógica capaz de lidar com propriedades de fases quânticas, evoluções unitárias, e entrelaçamento em sistemas compostos, uma vez que é um redesenhar da já existente LQP, [14], sobre uma representação de estados quânticos restrita a um espaço B munido de apenas duas bases computacionais, standard e Hadamard. Como instâncias de aplicação da lógica PhLQP, há uma prova formal para a correção do protocolo de Teletransporte Quântico, para o protocolo QLE para uma party quer de 2 quer de 4 agentes, e para o operador de QFT de 1, 2, e 3 qubits. Numa segunda fase, PhLQP é extendida com a conectiva ◦, conhecida como operador de consistência, uma conectiva característica das LFIs, [8, 21, 22]. E a partir desta conectiva a definição de estado quântico consistente é um conjunto de axiomas paraconsistentes próprios para o domínio quântico, FParQAxs. Um exemplo de aplicação da PhLQP◦ é a possibilidade de expressar e permitir correção para o comportamento da gate quântica universal, a Deutsch-gate