The Logic of Quantum Measurements

We apply our previously developed formalism of contexts of histories, suitable to deal with quantum properties at different times, to the measurement process. We explore the logical implications which are allowed by the quantum theory, about the realization of properties of the microscopic measured system, before and after the measurement process with a given pointer value.     

Quantum Circuit Synthesis using a New Quantum Logic Gate Library of NCV Quantum Gates

Since Controlled-Square-Root-of-NOT (CV, CV^?) gates are not permutative quantum gates, many existing methods cannot effectively synthesize optimal 3-qubit circuits directly using the NOT, CNOT, Controlled-Square-Root-of-NOT quantum gate library (NCV), and the key of effective methods is the mapping of NCV gates to four-valued quantum gates. Firstly, we use NCV gates to create the new quantum logic gate library, which can be directly used to get the solutions with smaller quantum costs efficiently. Further, we present a novel generic method which quickly and directly constructs this new optimal quantum logic gate library using CNOT and Controlled-Square-Root-of-NOT gates. Finally, we present several encouraging experiments using these new permutative gates, and give a careful analysis of the method, which introduces a new idea to quantum circuit synthesis.     

Quantum Logic for Quantum Computers

The following results obtained within a project of finding the algebra of statesin a general-purpose quantum computer are reported: (1) All operations of anorthomodular lattice, including the identity, are fivefold-defined; (2) there arenonorthomodular models for both quantum and classical logics; (3) there is afour-variable orthoarguesian lattice condition which contains all known orthoarguesianlattice conditions including six- and five-variable ones. Repercussions to quantumcomputers operating as quantum simulators are discussed.     

The Paraconsistent Logic of Quantum Superpositions

Physical superpositions exist both in classical and in quantum physics. However, what is exactly meant by ‘superposition’ in each case is extremely different. In this paper we discuss some of the multiple interpretations which exist in the literature regarding superpositions in quantum mechanics. We argue that all these interpretations have something in common: they all attempt to avoid ‘contradiction’. We argue in this paper, in favor of the importance of developing a new interpretation of superpositions which takes into account contradiction, as a key element of the formal structure of the theory, “right from the start”. In order to show the feasibility of our interpretational project we present an outline of a paraconsistent approach to quantum superpositions which attempts to account for the contradictory properties present in general within quantum superpositions. This approach must not be understood as a closed formal and conceptual scheme but rather as a first step towards a different type of understanding regarding quantum superpositions.     

Quantum Computational Logic

A quantum computational logic is constructed by employing density operators on spaces of qubits and quantum gates represented by unitary operators. It is shown that this quantum computational logic is isomorphic to the basic sequential effect algebra [0, 1].     

Typed Quantum Logic

The aim of this paper was to lift traditional quantum logic to its higher order version with the help of a type-theoretic method. A higher order axiomatic system is defined explicitly and then a sound and complete class of models is given. This is an attempt to provide a quantum counterpart of classical set theory or intuitionistic topos.     

Convex Quantum Logic

In this work we study the convex set of quantum states from a quantum logical point of view. We consider an algebraic structure based on the convex subsets of this set. The relationship of this algebraic structure with the lattice of propositions of quantum logic is shown. This new structure is suitable for the study of compound systems and shows new differences between quantum and classical mechanics. These differences are linked to the nontrivial correlations which appear when quantum systems interact. They are reflected in the new propositional structure, and do not have a classical analogue. This approach is also suitable for an algebraic characterization of entanglement and it provides a new entanglement criteria.     

Covariance and Quantum Logic

Considering the fundamental role symmetry plays throughout physics, it is remarkable how little attention has been paid to it in the quantum-logical literature. In this paper, we discuss G-test spaces—that is, test spaces hosting an action by a group G—and their logics. The focus is on G-test spaces having strong homogeneity properties. After establishing some general results and exhibiting various specimens (some of them exotic), we show that a sufficiently symmetric G-test space having an invariant, separating set of states with affine dimension n, is always representable in terms of a real Hilbert space of dimension n+1, in such a way that orthogonal outcomes are represented by orthogonal unit vectors.     

Regularity in Quantum Logic

In 1996, Harding showed that the binarydecompositions of any algebraic, relational, ortopological structure X form an orthomodular poset FactX. Here, we begin an investigation of the structuralproperties of such orthomodular posets of decompositions.We show that a finite set S of binary decompositions inFact X is compatible if and only if all the binarydecompositions in S can be built from a common n-arydecomposition of X. This characterization ofcompatibility is used to show that for any algebraic,relational, or topological structure X, the orthomodularposet Fact X is regular. Special cases of this result include the known facts that theorthomodular posets of splitting subspaces of an innerproduct space are regular, and that the orthomodularposets constructed from the idempotents of a ring are regular. This result also establishes theregularity of the orthomodular posets that Mushtariconstructs from bounded modular lattices, theorthomodular posets one constructs from the subgroups ofa group, and the orthomodular posets oneconstructs from a normed group with operators. Moreover,all these orthomodular posets are regular for the samereason. The characterization of compatibility is also used to show that for any structure X, thefinite Boolean subalgebras of Fact X correspond tofinitary direct product decompositions of the structureX. For algebraic and relational structures X, this result is extended to show that the Booleansubalgebras of Fact X correspond to representations ofthe structure X as the global sections of a sheaf ofstructures over a Boolean space. The above results can be given a physical interpretation as well.Assume that the true or false questions of a quantum mechanical system correspond tobinary direct product decompositions of the state spaceof the system, as is the case with the usual von Neumanninterpretation of quantum mechanics. Suppose S is asubset of . Then a necessary andsufficient condition that all questions in S can beanswered simultaneously is that any two questions in S can be answeredsimultaneously. Thus, regularity in quantum mechanicsfollows from the assumption that questions correspond todecompositions.     

The Power of Qutrit Logic for Quantum Computation

The critical merits acquired from quantum computation require running in parallel, which cannot be benefited from previous multi-level extensions and are exact our purposes. In this paper, with qutrit subsystems the general quantum computation further reduces into qutrit gates or its controlled operations. This extension plays parallizable and integrable with same construction independent of the qutrit numbers. The qutrit swapping as its basic operations for controlling can be integrated into quantum computers with present physical techniques. Our generalizations are free of elevating the system spaces, and feasible for the universal computation.     

Quantum Logic Network for Probabilistic Cloning Quantum States

We construct efficient quantum logic network for probabilistic cloning the quantum states used in implemented tasks for which cloning provides some enhancement in performance.     

Coreflections in Algebraic Quantum Logic

Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.     

Classical Limit and Quantum Logic

The analysis of the classical limit of quantum mechanics usually focuses on the state of the system. The general idea is to explain the disappearance of the interference terms of quantum states appealing to the decoherence process induced by the environment. However, in these approaches it is not explained how the structure of quantum properties becomes classical. In this paper, we consider the classical limit from a different perspective. We consider the set of properties of a quantum system and we study the quantum-to-classical transition of its logical structure. The aim is to open the door to a new study based on dynamical logics, that is, logics that change over time. In particular, we appeal to the notion of hybrid logics to describe semiclassical systems. Moreover, we consider systems with many characteristic decoherence times, whose sublattices of properties become distributive at different times.     

Quantum Logic and Non-Commutative Geometry

We propose a general scheme for the “logic” of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non-Commutative Geometry. It involves Baire^*-algebras, the non-commutative version of measurable functions, arising as envelope of the C ^*-algebras identifying the topology of the (non-commutative) phase space. We outline some consequences of this proposal in different physical systems. This approach in particular avoids some problematic features appearing in the definition of physical states in the standard (W ^*-)algebraic approach to classical mechanics.     

A Link between Quantum Logic and Categorical Quantum Mechanics

Abramsky and Coecke (Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425, IEEE Comput. Soc., New York, 2004) have recently introduced an approach to finite dimensional quantum mechanics based on strongly compact closed categories with biproducts. In this note it is shown that the projections of any object A in such a category form an orthoalgebra Proj  A. Sufficient conditions are given to ensure this orthoalgebra is an orthomodular poset. A notion of a preparation for such an object is given by Abramsky and Coecke, and it is shown that each preparation induces a finitely additive map from Proj  A to the unit interval of the semiring of scalars for this category. The tensor product for the category is shown to induce an orthoalgebra bimorphism Proj  A×Proj  B→Proj (A ⊗ B) that shares some of the properties required of a tensor product of orthoalgebras. These results are established in a setting more general than that of strongly compact closed categories. Many are valid in dagger biproduct categories, others require also a symmetric monoidal tensor compatible with the dagger and biproducts. Examples are considered for several familiar strongly compact closed categories.     

量子态的等价类与量子逻辑门

在量子力学中,态的演化是一个幺正演化过程,态的演化过程可以用演化算子对态的作用来表示,幺正演化过程是时间可逆的.基于这一基本事实,Gerard't Hoofl引进了量子态的等价类概念,并用两组等价类之间的变换来描述量子态的幺正演化.本文利用等价类的概念及其变换来探究构建量子信息论中常用的通用量子门,给出通用量子门的推广形式.最后说明这些通用量子门可以基于双qubit体系内在的相互作用Hamilton量得以实现.