In the first chapter we introduce new forms of quantum logic suggested by quantum computation, 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, representing a possible pure state of a compound quantum system. The generalization to mixed states might be useful to analyse entanglement-phenomena. We study structural properties of density operators systems, where some basic quantum logical gate are defined. We introduce the notions of standard reversible and standard irreversible quantum computational structure. Quantum computational logics represent non standard examples of unsharp quantum logic, where the non-contradiction principle is violated, while conjunctions and disjunctions are strongly non-idempotent. In this framework, any sentence of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister associated to the atomic subformulas of the formula into the quregister associated to the formula. We generalize the quantum computational semantics in order to represent some typical quantum holistic situations where the meaning of the whole determines the contextual meanings of the parts, but not vice versa. We describe some holistic models in the framework of Mach-Zehnder interferometers. In the second chapter we extend the basic principles and results of conservative logic to include the main features of many-valued logics with a finite number of truth values. Different approaches to many-valued logics are examined in order to determine some possible functionally complete sets of logic connectives. In particular, we consider the typical connectives of Lukasiewicz and Gödel logics, as well as Zawirski/Chang's MV-algebras. As a result, we describe some possible three-valued and finite-valued universal gates which realize a functionally complete set of fundamental connectives. One of the purposes of this work is to show that the framework of reversible and conservative computation can be extended toward some non classical "reasoning environments", originally proposed to deal with propositions which embed imprecise and uncertain information, that are usually based upon many-valued and modal logics. We also describe a possible quantum realization of the proposed gates, using creation and annihilation operators. These formulas are obtained using three techniques: a "brute force" technique, an extension of the Conditional Quantum Control method introduced by Barenco, Deutsch, Ekert and Jozsa in 1995, and a new technique that we call the Constants method. The formulas obtained with these techniques are sums of local operators. In the brute force technique, each local operator corresponds to a single row of the truth table of the implemented gate. In the Conditional Quantum Control method, an assumption is made on the behavior of the gate: this assumption usually leads to much shorter formulas, but on the other hand the method cannot be applied to the gates that do not satisfy it. Our Constants method is a general technique, since we do not impose any constraint on the structure or on the behavior of the gate; the length of the corresponding formula is minimized by looking at all possible realizations of the desired connectives, obtained by setting some input lines of the gate to appropriate constant values. In the third chapter we discuss a model to realize the Petri-Fredkin gate by a Mach-Zehnder like optical device with a non-linear component. The device uses optical nonlinear Kerr effect generated into a substance with an intensity dependent refractive index: the intensity dependence of the refractive index is the source of nonlinearity.

Titolo: | Logics from Quantum Information |

Tutti gli autori: | LEPORINI, ROBERTO |

Data di pubblicazione: | 2005 |

Abstract (ita): | In the first chapter we introduce new forms of quantum logic suggested by quantum computation, 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, representing a possible pure state of a compound quantum system. The generalization to mixed states might be useful to analyse entanglement-phenomena. We study structural properties of density operators systems, where some basic quantum logical gate are defined. We introduce the notions of standard reversible and standard irreversible quantum computational structure. Quantum computational logics represent non standard examples of unsharp quantum logic, where the non-contradiction principle is violated, while conjunctions and disjunctions are strongly non-idempotent. In this framework, any sentence of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister associated to the atomic subformulas of the formula into the quregister associated to the formula. We generalize the quantum computational semantics in order to represent some typical quantum holistic situations where the meaning of the whole determines the contextual meanings of the parts, but not vice versa. We describe some holistic models in the framework of Mach-Zehnder interferometers. In the second chapter we extend the basic principles and results of conservative logic to include the main features of many-valued logics with a finite number of truth values. Different approaches to many-valued logics are examined in order to determine some possible functionally complete sets of logic connectives. In particular, we consider the typical connectives of Lukasiewicz and Gödel logics, as well as Zawirski/Chang's MV-algebras. As a result, we describe some possible three-valued and finite-valued universal gates which realize a functionally complete set of fundamental connectives. One of the purposes of this work is to show that the framework of reversible and conservative computation can be extended toward some non classical "reasoning environments", originally proposed to deal with propositions which embed imprecise and uncertain information, that are usually based upon many-valued and modal logics. We also describe a possible quantum realization of the proposed gates, using creation and annihilation operators. These formulas are obtained using three techniques: a "brute force" technique, an extension of the Conditional Quantum Control method introduced by Barenco, Deutsch, Ekert and Jozsa in 1995, and a new technique that we call the Constants method. The formulas obtained with these techniques are sums of local operators. In the brute force technique, each local operator corresponds to a single row of the truth table of the implemented gate. In the Conditional Quantum Control method, an assumption is made on the behavior of the gate: this assumption usually leads to much shorter formulas, but on the other hand the method cannot be applied to the gates that do not satisfy it. Our Constants method is a general technique, since we do not impose any constraint on the structure or on the behavior of the gate; the length of the corresponding formula is minimized by looking at all possible realizations of the desired connectives, obtained by setting some input lines of the gate to appropriate constant values. In the third chapter we discuss a model to realize the Petri-Fredkin gate by a Mach-Zehnder like optical device with a non-linear component. The device uses optical nonlinear Kerr effect generated into a substance with an intensity dependent refractive index: the intensity dependence of the refractive index is the source of nonlinearity. |

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