相干与信息守恒及其在Mach-Zehnder干涉中的应用

自量子力学诞生以来,相干性和互补性一直是被广泛而深入研究的两个重要课题.随着量子信息近年来的发展,人们引入了若干度量来定量地刻画相干性和互补性.本文建立两个信息守恒关系式,分别基于"Bures距离-保真度"和"对称-非对称",并且利用它们来刻画相干性和互补性.具体来说,首先从信息守恒的观点解释Bures距离和保真度的互补关系,并由此自然推导出Mach-Zehnder干涉仪中的Englert"干涉-路径"互补关系.其次在量子态和信道相互作用的一般框架中讨论"对称-非对称"信息守恒关系,并揭示其与Bohr互补性和量子相干性的内在联系.最后,在Mach-Zehnder干涉仪中探讨相干、退相干及互补性,刻画两个信息守恒关系之间的密切联系.     

Hilbertian quantum theory as the theory of complementarity

It is demonstrated that the notion of complementary physical quantities assumes the possibility of performing ideal first-kind measurements of such quantities. This then leads to an axiomatic reconstruction of the Hilbertian quantum theory based on the complementarity principle and on its connection with the measurement theoretical idealization known as the projection postulate. As the notion of complementary physical quantities does not presuppose the notion of probability, the given axiomatic reconstruction reveals complementarity as an essential reason for the irreducibly probabilistic nature of the quantum theory.     

Fundamental principles of quantum theory. II. From a convexity scheme to the DHB theory

Some classical and quantum theories are characterized within the convexity approach to probabilistic physical theories. In particular, the structure of the so-called DHB quantum theory will be analyzed. It turns out that the natural generalization of the standard Hubert space quantum mechanics, the operational one, is such a theory. The operational Hilbert space quantum theory will be reconstructed from the (weak) projection postulate and the complementarity principle. This is then used to argue that the DHB quantum theory is identical with the operational Hilbert space quantum theory.     

From wave-particle duality to wave-particle-mixedness triality: an uncertainty approach

The wave-particle duality, as a manifestation of Bohr’s complementarity, is usually quantified in terms of path predictability and interference visibility. Various characterizations of the wave-particle duality have been proposed from an operational perspective, most of them are in forms of inequalities, and some of them are expressed in forms of equalities by incorporating entanglement or coherence. In this work, we shed different insights into the nature of the wave-particle duality by casting it into a form of information conservation in a multi-path interferometer, with uncertainty as a unified theme. More specifically, by employing the simple yet fundamental concept of variance, we establish a resolution of unity, which can be interpreted as a complementarity relation among wave feature, particle feature, and mixedness of a quantum state. This refines or reinterprets some conventional approaches to wave-particle duality, and highlights informational aspects of the issue. The key idea of our approach lies in that a quantum state, as a Hermitian operator, can also be naturally regarded as an observable, with measurement uncertainty (in a state) and state uncertainty (in a measurement) being exploited to quantify particle feature and wave feature of a quantum state, respectively. These two kinds of uncertainties, although both are defined via variance, have fundamentally different properties and capture different features of a state. Together with the mixedness, which is a kind of uncertainty intrinsic to a quantum state, they add up to unity, and thus lead to a characterization of the wave-particle-mixedness complementarity. This triality relation is further illustrated by examples and compared with some popular wave-particle duality or triality relations.     

Uncertainty and complementarity in axiomatic quantum mechanics

In this work an investigation of the uncertainty principle and the complementarity principle is carried through. A study of the physical content of these principles and their representation in the conventional Hilbert space formulation of quantum mechanics forms a natural starting point for this analysis. Thereafter is presented more general axiomatic framework for quantum mechanics, namely, a probability function formulation of the theory. In this general framework two extra axioms are stated, reflecting the ideas of the uncertainty principle and the complementarity principle, respectively. The quantal features of these axioms are explicated. The sufficiency of the state system guarantees that the observables satisfying the uncertainty principle are unbounded and noncompatible. The complementarity principle implies a non-Boolean proposition structure for the theory. Moreover, nonconstant complementary observables are always noncompatible. The uncertainty principle and the complementarity principle, as formulated in this work, are mutually independent. Some order is thus brought into the confused discussion about the interrelations of these two important principles. A comparison of the present formulations of the uncertainty principle and the complementarity principle with the Jauch formulation of the superposition principle is also given. The mutual independence of the three fundamental principles of the quantum theory is hereby revealed.     

Uncertainty relations for quantum coherence with respect to mutually unbiased bases

The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.     

Some basic principles in dynamic theory of viscoelastic materials with voids

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified way proposed by Luo, some basic principles in the dynamic theory of viscoelastic materials with voids can be estab- lished systematically. In this paper, an important integral relation in terms of con- volutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem, but also to derive systemati- cally the complementary functionals for the eight-field, six-field, four-field simpli- fied Gurtin-type variational principles and the potential energy-functional for the two-field one in the dynamic theory of viscoelastic materials with voids by the generalized Legendre transformations given in this paper. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly.     

Some basic principles for the linear theory of piezoelectric micropolar elastodynamics

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,in a simple and unified way proposed by Luo,some basic principles in the linear theory of piezoelectric micropolar elastodynamics can be established systematically. In this paper,an important integral relation in terms of convolutions is given,which can be considered as the generalized principle of virtual work in mechanics. Based on this relation,it is not only possible to obtain the principle of virtual work and the reciprocal theorem,but also to systematically derive the complementary functionals for the eleven-field,nine-field and six-field simplified Gurtin-type variational principles and the potential energy-functional for the three-field one in the linear theory of piezoelectric micropolar elastodynamics by the generalized Legendre transformations given in this paper. Furthermore,with this approach,the intrinsic relationships among various principles can be explained clearly.     

Robustness of interferometric complementarity under decoherence

Interferometric complementarity is known to be one of the most nonclassical manifestations of the quantum formalism. It is commonly known as wave-particle duality and has been studied presently from the perspective of quantum information theory where wave and particle nature of a quantum system, called quanton, are characterised by coherence and path distinguishability respectively. We here consider the effect of noisy detectors on the complementarity relation. We report that by suitably choosing the initial quanton and the detector states along with the proper interactions between the quanton and the detectors, one can reduce the influence of noisy environment on complementarity, thereby pushing it towards saturation. To demonstrate this, three kinds of noise on detectors and their roles on the saturation of the complementarity relation are extensively studied. We also observe that for fixed values of parameters involved in the process, asymmetric quanton state posses low value of coherence while it can have a higher amount of distinguishability, and hence it has the potential to enhance the duality relation.     

Nonlinear classical theory of electromagnetism

A topological theory of electric charge is given. Einstein's criteria for the completion of classical electromagnetic theory are summarized and their relation to quantum theory and the principle of complementarity is indicated. The inhibiting effect that this principle has had on the development of physical thought is discussed. Developments in the theory of functions on nonlinear spaces provide the conceptual framework required for the completion of electromagnetism. The theory is based on an underlying field which is a continuous mapping of space-time into points on the two-sphere.     

Is the Devil in h?

This note is a part of my effort to rid quantum mechanics (QM) nonlocality. Quantum nonlocality is a two faced Janus: one face is a genuine quantum mechanical nonlocality (defined by the Lüders’ projection postulate). Another face is the nonlocality of the hidden variables model that was invented by Bell. This paper is devoted the deconstruction of the latter. The main casualty of Bell’s model is that it straightforwardly contradicts Heisenberg’s uncertainty and Bohr’s complementarity principles generally. Thus, we do not criticize the derivation or interpretation of the Bell inequality (as was done by numerous authors). Our critique is directed against the model as such. The original Einstein-Podolsky-Rosen (EPR) argument assumed the Heisenberg’s principle without questioning its validity. Hence, the arguments of EPR and Bell differ crucially, and it is necessary to establish the physical ground of the aforementioned principles. This is the quantum postulate: the existence of an indivisible quantum of action given by the Planck constant. Bell’s approach with hidden variables implicitly implies rejection of the quantum postulate, since the latter is the basis of the reference principles.     

Quantum mechanics and the concept of joint probability

The concepts of joint probability as implied by the Copenhagen and realist interpretations of quantum mechanics are examined in relation to (a) the rules for manipulation of probabilistic quantities, and (b) the role of the Bell inequalities in assessing the completeness of standard quantum theory. Proponents of completeness of the Copenhagen interpretation are required to accept a modification of the classical laws of probability to provide a mechanism for complementarity. A new formulation of the locality postulate is given, not involving factorizability or conditional probabilities, which clarifies the role of locality in the derivation of the Bell inequalities.     

Coherence migration in high-dimensional bipartite systems

The conservation law for first-order coherence and mutual correlation of a bipartite qubit state was firstly proposed by Svozilík et al., and their theories laid the foundation for the study of coherence migration under unitary transformations. In this paper, we generalize the framework of first-order coherence and mutual correlation to an arbitrary (m $\otimes$ n)-dimensional bipartite composite state by introducing an extended Bloch decomposition form of the state. We also generalize two kinds of unitary operators in high-dimensional systems, which can bring about coherence migration and help to obtain the maximum or minimum first-order coherence. Meanwhile, the coherence migration in open quantum systems is investigated. We take depolarizing channels as examples and establish that the reduced first-order coherence of the principal system over time is completely transformed into mutual correlation of the (2 $\otimes$ 4)-dimensional system-environment bipartite composite state. It is expected that our results may provide a valuable idea or method for controlling the quantum resource such as coherence and quantum correlations.     

A wave function for stock market returns

The instantaneous return on the Financial Times-Stock Exchange (FTSE) All Share Index is viewed as a frictionless particle moving in a one-dimensional square well but where there is a non-trivial probability of the particle tunneling into the well’s retaining walls. Our analysis demonstrates how the complementarity principle from quantum mechanics applies to stock market prices and of how the wave function presented by it leads to a probability density which exhibits strong compatibility with returns earned on the FTSE All Share Index. In particular, our analysis shows that the probability density for stock market returns is highly leptokurtic with slight (though not significant) negative skewness. Moreover, the moments of the probability density determined under the complementarity principle employed here are all convergent — in contrast to many of the probability density functions on which the received theory of finance is based.     

Transformation relation between coherence and entanglement for two-qubit states

Entanglement and coherence are two important resources in quantum information theory. A question naturally arises: Is there some connection between them? We prove that the entanglement of formation and the first-order coherence of two-qubit states satisfy an inequality relation. Two-qubit pure state reaches the upper bound of this inequality. A large number of randomly generated states are used to intuitively verify the complementarity between the entanglement of formation and the first-order coherence. We give the maximum accessible coherence of two-qubit states. Our research results will provide a reliable theoretical basis for conversion of the two quantum resources.     

On the Wootters-Zurek development of Einstein's two-slit experiment

We consider the compatibility of the Wootters and Zurek development of information theory as applied to the two-slit experiment with the principle of complementarity. We also consider the limitations of aspects of Wootters and Zurek's analysis, and, independently of complementarity, the extent to which Wootters and Zurek's information theory can be considered a fundamental interpretation of the quantum theory (as applied to particle-wave duality). The question of particle-wave uncertainty relations will also be taken up.     

A new theory of elementary matter

This is the first of a series of articles that reviews and expands upon a new theory of elementary matter. This paper presents an exposition of the philosophical approach and its general implications. The ensuing explicit form of the mathematical expression of the theory and several applications in the atomic and elementary particle domains will be developed in the succeeding parts of this series.The theory is based on three axioms: the principle of general relativity, a generalized Mach principle, and a correspondence principle. The approach is basically a deterministic, relativistic field theory which fully incorporates the idea that any realistic physical system is in facta closed system, without separable parts. It is shown that the most primitive mathematical expression of this theory, following as anecessary consequence of its axioms, is in terms of a set of coupled nonlinear spinor field equations. Nevertheless, the exact formalism is constructed to asymptotically approach the quantum mechanical formalism for a many-particle system, in the limit of sufficiently small energy-momentum transfer among the components of the considered closed system. Thus, all of the mathematical predictions of nonrelativistic quantum mechanics are contained in this theory, as a mathematical approximation. However, predictions follow from the exact form of this theory (where energy-momentum transfer can be arbitrarily large) that are not contained in the quantum theory.