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Entropy

Holistic Type Extension for Classical Logic via Toffoli Quantum Gate

Héctor Freytes, Roberto Giuntini, Giuseppe Sergioli

Published Jun 27, 2019• 2 citations
Publisher

Abstract

A holistic extension of classical propositional logic is introduced via Toffoli quantum gate. This extension is based on the framework of the so-called "quantum computation with mixed states", where also irreversible transformations are taken into account. Formal aspects of this new logical system are detailed: in particular, the concepts of tautology and contradiction are investigated in this extension. These concepts turn out to receive substantial changes due to the non-separability of some quantum states; as an example, Werner states emerge as particular cases of "holistic" contradiction.

DOI
10.3390/e21070636

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A holistic extension of classical propositional logic is introduced via Toffoli quantum gate. This extension is based on the framework of the so-called "quantum computation with mixed states", where also irreversible transformations are taken into account. Formal aspects of this new logical system are detailed: in particular, the concepts of tautology and contradiction are investigated in this extension. These concepts turn out to receive substantial changes due to the non-separability of some quantum states; as an example, Werner states emerge as particular cases of "holistic" contradiction.
Does quantum holism give rise to new forms of abstract holistic semantics? This question can be discussed in the framework of quantum computational logics, non-standard versions of quantum logic that have been suggested by the theory of logical gates in quantum computation. The basic idea of the quantum computational semantics is the following: meanings of sentences are identified with quantum information quantities, represented by pure or mixed states of special quantum systems. In particular cases, the meaning of a sentence may correspond to an entangled state. On this basis, one can formally describe some typical holistic semantic situations, where the meaning of the whole determines the meanings of its parts, but generally not vice versa.
The theory of gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics, where the meaning of a sentence is identified with a system of qubits in a pure or, more generally, mixed state. In this framework, any formula of the language gives rise to a quantum circuit that transforms the state associated to the atomic subformulas into the state associated to the formula and vice versa. On this basis, some holistic semantic situations can be described, where the meaning of whole determines the meaning of the parts, by non-linear and anti-unitary operators. We prove that the semantics with such operators and the semantics with unitary operators turn out to characterize the same logic.
The theory of gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics, where the meaning of a sentence is identified with a system of qubits in a pure or, more generally, mixed state. In this framework, any formula of the language gives rise to a quantum circuit that transforms the state associated to the atomic subformulas into the state associated to the formula and vice versa. On this bases, some holistic semantic situations can be described, where the meaning of whole determine the meaning of the parts, by non-linear and anti-unitary operators. We prove that the semantics with such operators and the semantics with unitary operators turn out to characterize the same logic.
The research on many valued interpretation in quantum computing [1,3] is rather recent and the theory is not fully developed yet. The main advantage of this approach seems to be the complexity gain when a computation is based upon general quantum digits (usually called qudits) if compared with time spent by the same algorithm, when running in a equivalent system built up by means of qubits (binary digits). If this assumption is sound, then temporal complexity of algorithms could be significantly simplified. The use of many-valued semantics in classical logic doesn't convey any precise idea of its physical application, as it happens in the framework of quantum gates. However, we propose an analysis motivated by the known use of finite-many-valued semantic in non-quantum logics. We shall produce a propositional representation for classical gates. We make explicit the usually implicit relationship between classical and quantum logics, extend in a natural way the many-valued semantics, as used in axiomatic systems of classical logical, to the quantum case (by using Toffoli gate and algebraic properties), define many-valued digits (qu-d-its) and reversible gates based upon them, and extend the functional completeness (“universality”) of some gates to the new situation. We conjecture the possibility of enhancing the performance of quantum algorithms, when running in a many-valued quantum system.