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International Journal of Quantum Information

LOGICS FROM QUANTUM COMPUTATION WITH BOUNDED ADDITIVE OPERATORS

Cesarino Bertini, Roberto Leporini

Published Apr 1, 2012• 3 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749912500360

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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 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.
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 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 α is identified with a quantum information quantity, represented by a density operator of a Hilbert space, whose dimension depends on the logical complexity of α. At the same time, the logical connectives of the language are interpreted as operations defined in terms of quantum logical gates. Standard quantum computational models can be described as special cases of Fock space models, where the meaning of any sentence is localized in a precise sector of a Fock space ℱ. From an intuitive point of view, the increasing number of particles described in the different sectors of ℱ can be interpreted as increasing information.
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 quantum information quantity, represented by a quregister (a system of qudits) or, more generally, by a mixture of quregisters (called qumix), whose dimension depends on the logical complexity of the sentence. At the same time, the logical connectives are interpreted as logical operations defined in terms of quantum logical gates. Physical models of quantum computational logics can be built by means of Mach-Zehnder interferometers.
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 in a pure state) or, more generally, with a mixture of quregisters (called qumix). Following an approach proposed by Domenech and Freytes, we apply residuated structures associated with fuzzy logic to develop certain aspects of information processing in quantum computing from a logical perspective. For this purpose, we introduce an axiomatic system whose natural interpretation is the irreversible quantum Poincaré algebra. Such a system allows to establish a completeness theorem for the treatment of quantum information.