Quantum Computational Logics and Possible Applications

In quantum computational logics meanings of formulas are identified with quantum information quantities: systems of qubits or, more generally, mixtures of systems of qubits. We consider two kinds of quantum computational semantics: (1) a compositional semantics, where the meaning of a compound formula is determined by the meanings of its parts; (2) a holistic semantics, which makes essential use of the characteristic “holistic” features of the quantum-theoretic formalism. The compositional and the holistic semantics turn out to characterize the same logic. In this framework, one can introduce the notion of quantum-classical truth table, which corresponds to the most natural way for a quantum computer to calculate classical tautologies. Quantum computational logics can be applied to investigate different kinds of semantic phenomena where holistic, contextual and gestaltic patterns play an essential role (from natural languages to musical compositions).     

The Algebraic Structure of an Approximately Universal System of Quantum Computational Gates

Shi and Aharonov have shown that the Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. We study the basic algebraic properties of this system by introducing the notion of Shi-Aharonov quantum computational structure. We show that the quotient of this structure is isomorphic to a structure based on a particular set of complex numbers (the closed disc with center and radius ). Dedicated to Pekka Lahti.     

Automata Theory Based on Quantum Logic: Recognizability and Accessibility

Inspired by Ying’s work on automata theory based on quantum logic and classical automata theory, we introduce the concepts of reversal, accessible, coaccessible and complete part of finite state automata based on quantum logic. Some properties of them are discussed. More importantly we investigate the recognizability and accessibility properties of these types on the framework of quantum logic by employing the approach of semantic analysis. Foundation: supported by the National Natural Science Foundation of China (No. 10671030).     

An Operational Characterization for Optimal Observation of Potential Properties in Quantum Theory and Signal Analysis

Quantum logic introduced a paradigm shift in the axiomatization of quantum theory by looking directly at the structural relations between the closed subspaces of the Hilbert space of a system. The one dimensional closed subspaces correspond to testable properties of the system, forging an operational link between theory and experiment. Thus a property is called actual, if the corresponding test yields “yes” with certainty. We argue a truly operational definition should include a quantitative criterion that tells us when we ought to be satisfied that the test yields “yes” with certainty. This question becomes particularly pressing when we inquire how the usual definition can be extended to cover potential, rather than actual properties. We present a statistically operational candidate for such an extension and show that its representation automatically captures some essential Hilbert space structure. If it is the nature of observation that is responsible for the Hilbert space structure, then we should be able to give examples of theories with scope outside the domain of quantum theory, that employ its basic structure, and that describe the optimal extraction of information. We argue signal analysis is such an example. This work was supported by the Flemish Fund for Scientific Research (FWO) project G.0362.03N.     

单光子纠缠态的纠缠转移和量子隐形传态

使用光学分束器和单光子源,利用单光子态和真空态制备出了纠缠单光子态.利用光学分束器作用和单光子探测,实现了三个通讯伙伴之间的纠缠转移.提出了一个关于纠缠单光子态的量子隐形传态方案.在这个方案中,被传送的是一个未知的单光子纠缠态.通讯双方使用的量子信道是两个单光子纠缠态.通过使用分束器作用和对输出态进行光子测量以及在经典信息的帮助下,纠缠转移和量子隐形传态的过程被完成.     

Intrinsic Properties of Quantum Systems

A new realist interpretation of quantum mechanics is introduced. Quantum systems are shown to have two kinds of properties: the usual ones described by values of quantum observables, which are called extrinsic, and those that can be attributed to individual quantum systems without violating standard quantum mechanics, which are called intrinsic. The intrinsic properties are classified into structural and conditional. A systematic and self-consistent account is given. Much more statements become meaningful than any version of Copenhagen interpretation would allow. A new approach to classical properties and measurement problem is suggested. A quantum definition of classical states is proposed.     

双光子过程耦合腔系统中光场的量子特性

本文研究了双光子过程原子与耦合腔相互作用系统中腔场的压缩效应和反聚束效应.考虑系统总激发数等于2的情况,利用数值计算方法讨论了腔场间的耦合强度变化和原子与腔场间耦合强度变化对反聚束效应的影响.研究结果表明:腔场不呈现出压缩效应;腔场的反聚束效应与原子与腔场的耦合系数之间,以及与腔场间的耦合系数之间都存在着非线性关系.     

共振光散射的时间量子理论

本文发展了Ｓｃｈｏｍｍｅｒｓ的时间观点，定义了时间，尤其是表征量子系统光散射的散射时间和共振散射时间。通过散射时间本征态的假定实现了散射时间的量子化，得到了量子化的共振散射时间。对原子的弹性光散射和Ｒａｍａｎ散射的成功应用推出了原子和原子的价电子逐级电离所形成的离子的所有原子能级的普适近似公式。     

双光子过程耦合腔系统中光场的量子特性

本文研究了双光子过程原子与耦合腔相互作用系统中腔场的压缩效应和反聚束效应.考虑系统总激发数等于2的情况,利用数值计算方法讨论了腔场间的耦合强度变化和原子与腔场间耦合强度变化对反聚束效应的影响.研究结果表明:腔场不呈现出压缩效应;腔场的反聚束效应与原子与腔场的耦合系数之间,以及与腔场间的耦合系数之间都存在着非线性关系,     

用腔场中的二能级势阱离子实现量子逻辑门

利用光腔中的势阱粒子同时与外激光场和腔场发生相互作用的特性，我们提出了一种量子逻辑门的实现方案。在该方案中，我们采用文献[10-12]中的模型。文献[11-12]中实现的逻辑门是以离子内态和运动态作为量子比特，腔态充当辅助比特在计算过程中保持在基态。而[10]要求离子内态保持为基态，利用离子运动态和腔态构成量子比特。与文献[10-12]不同的是，我们实现的量子逻辑门是以粒子内态和腔态作为比特，而势阱离子的运动态作为辅助比特始终保持在基态。而且，我们对该方案的实验要求进行了讨论。     

Intuitionistic Quantum Logic of an <Emphasis Type="Italic">n</Emphasis>-level System

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra M _n (?) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ _n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ?(M _n (?)) of projections in M _n (?) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice \(\mathcal{O}(\Sigma_{n})\) of functions from the poset \(\mathcal{C}(M_{n}(\mathbb{C}))\) of all unital commutative C*-subalgebras C of M _n (?) to ?(M _n (?)). The lattice \(\mathcal{O}(\Sigma_{n})\) is essentially the (pointfree) topology of the quantum phase space Σ _n , and as such defines a Heyting algebra. Each element of \(\mathcal{O}(\Sigma_{n})\) corresponds to a “Bohrified” proposition, in the sense that to each classical context \(C\in\mathcal{C}(M_{n}(\mathbb{C}))\) it associates a yes-no question (i.e. an element of the Boolean lattice ?(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.     

A Topological Study of Contextuality and Modality in Quantum Mechanics

Kochen–Specker theorem rules out the non-contextual assignment of values to physical magnitudes. Here we enrich the usual orthomodular structure of quantum mechanical propositions with modal operators. This enlargement allows to refer consistently to actual and possible properties of the system. By means of a topological argument, more precisely in terms of the existence of sections of sheaves, we give an extended version of Kochen–Specker theorem over this new structure. This allows us to prove that contextuality remains a central feature even in the enriched propositional system. Graciela Domenech is a fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Hector Freytes is a fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).     

四能级原子量子相干过程中瞬态光学特性

利用半经典理论研究了N-型四能级相干原子的瞬态光学特性。理论分析表明：当信号光开启的瞬间，介质对探针光的非线性吸收从接近于零的状态迅速增加，经过驰豫振荡过程，趋于稳定状态。本文的研究结果将为吸收型光子开关的设计提供理论依据和参考。     

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.     

与运动二能级原子作用的双模光场的量子特性

用全量子化理论,研究了与运动二能级原子相互作用的双模量子化光场的量子特性,分析了场模结构对平均光子数,单模二阶相干度,两模间的相关性的影响. 研究结果表明,随着场模结构参量的增大,平均光子数回复周期变短,振荡加剧,在崩塌区域内振荡出现分裂, 同时原子运动对光场的二阶相干性有明显的影响.     

Duality Quantum Computers and Quantum Operations

We present a mathematical theory for a new type of quantum computer called a duality quantum computer that is similar to one that has recently been proposed. We discuss the nonunitarity of certain circuits of a duality quantum computer. We then discuss the relevance of this work to quantum operations and their convexity theory. This discussion is based upon isomorphism theorems for completely positive maps.     

Time,Clocks and Parametric Invariance

In the context of a parametric theory (with the time being a dynamical variable) we consider here the coupling between the quantum vacuum and the background gravitation that pervades the universe (unavoidable because of the universality and long range of gravity). We show that this coupling, combined with the fourth Heisenberg relation, would break the parametric invariance of the gravitational equations, introducing thus a difference between the marches of the atomic and the astronomical clocks. More precisely, they would be progressively and adiabatically desynchronized with respect to one another in such a way that the latter would lag behind the former. This would produce a discrepancy between gravitational theory and observations, which use astronomical and atomic time respectively. It turns out that this result, surprising at it might be, is fully compatible with current physics, since it does not conflict with any known physical law or principle. We argue that this phenomenon must be studied, since it could have cosmological consequences.     

基于两节点间信息互传的量子特性

运用全量子理论,对腔耦合系统构建的两个节点,考虑节点内腔模与量子位(qubit)的耦合,结合数值计算,用熵表示信源的不确定性,对信源发出的信息进行度量,研究了两个节点相互进行信息传输过程的熵演化.通过两能级粒子与腔模的耦合强度、腔-腔之间的跃迁耦合系数和失谐量三个参数对熵变化进行分析,结论表明在共振条件下,节点间相互传输信息过程中,耦合双腔构成的两个节点熵呈现准周期性坍塌与复苏振荡变化特征,节点1与节点2熵的峰值交替出现;两个节点之间用跳跃频率λ/2π的光子作为信息传送的数椐总线,失谐使两个节点内量子位的一个频率高,一个频率低,无论先操纵哪一个量子位,在系统稳定工作状态下,量子态演化的信息传递方向总是从频率快的量子位向频率慢的量子位传递,此特性可扩展至多量子位之间量子信息的传递.用于两个节点间的远程操控.