Holistic and Compositional Logics Based on the Bertini Gate
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Holistic and Compositional Logics Based on the Bertini Gate Roberto Leporini1 Accepted: 2 September 2020 © Springer Nature B.V. 2020
Abstract The theory of logical gates in quantum computation has inspired the development of new forms of quantum logic where the meaning of a formula is identified with a density operator and the logical connectives are interpreted as operations defined in terms of quantum gates. We show some relations between the Bertini gate and many valued connectives by probability values. On this basis, one can deal with quantum circuits as expressions in an algebraic environment such as product many valued algebra for combinational circuits. As can be expected, we show that the compositional logic characterized by the qubit semantics is stronger than the compositional Łukasiewicz quantum computational logic by a counterexample. But, in the holistic case, we conjecture that they can characterize the same logic. Keywords Quantum logics · Quantum gates · Holism · Contextuality · Ambiguity
1 Introduction The mathematical formalism of quantum theory has inspired the development of quantum logics based on special classes of algebraic structures defined in a Hilbert-space environment. Interesting generalizations of quantum logic introduced by Birkhoff and von Neumann are the so called fuzzy (or unsharp) quantum logics that can be semantically characterized by referring to different classes of algebraic structures whose support is the set of all effects of a Hilbert space (Dalla Chiara et al. 2016). A different approach to quantum logic has been developed in the framework of quantum computational logics, inspired by the theory of quantum computation (Dalla Chiara et al. 2005). While sharp and unsharp quantum logics refer to possible structures of physical events, the basic objects of quantum computational logics are pieces of quantum information: possible states of quantum systems that can store the information in question. The simplest piece of quantum information is a qubit (state): a unit-vector of the Hilbert space ℂ2 that can be represented as a superposition �𝜓⟩ = c0 �0⟩ + c1 �1⟩ . The two elements of the canonical basis of ℂ2 , �0⟩ = (1, 0) and �1⟩ = (0, 1) , represent the classical bits or, equivalently, the two classical truth-values. It is interesting to consider a “many-valued * Roberto Leporini [email protected] 1
Dipartimento di Ingegneria Gestionale dell’Informazione e della Produzione, Università di Bergamo, viale Marconi 5, 24044 Dalmine, BG, Italy
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generalization” of qubits, represented by qudits: unit-vectors living in a space ℂd , where d ≥ 2 . In the many-valued generalization, one can deal with d truth values and consider many-valued connectives which allow one to obtain logical truths (Bertini and Leporini 2007) differently from the qubit case. The aim of this paper is to introduce compositional and holistic logics based on the Bertini gate in the qudit framework of quantum computation. The most natural semantics is a form of ho
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