First, based on the uncertainty principle in quantum theory and new generalized uncertainty principle, we research uncertain relations of entropy, information and thermodynamics, and propose the speed of light and some fundamental constants should be uncertain. Next, in the quantum entangled states, velocities of different phases may be superluminal, and obey possibly general Lorentz transformation (GLT). Third, we search its possible development in superstring, and the extensive quantum theory, various matters and different phases, etc. Fourth, the duality exists probably only under interactions. Some nonlinear quantum equations and wave equations have soliton solutions, which correspond to particles. This is mathematical description of wave-particle duality. Fifth, based on the soliton-chaos double solutions of nonlinear equations, we propose the field-particle duality, and field (wave)-quantum-chaos ternary. Sixth, the irreversibility and statistics are discussed, and we derive operators of entropy, etc. New measuring technology and some developed theories already challenge the uncertainty principle and duality, which may not be held under certain conditions, and must be modified and developed.
First, based on the uncertainty principle in quantum theory and new generalized uncertainty principle, we research uncertain relations of entropy, information and thermodynamics, and propose the speed of light and some fundamental constants should be uncertain. Next, in the quantum entangled states, velocities of different phases may be superluminal, and obey possibly general Lorentz transformation (GLT). Third, we search its possible development in superstring, and the extensive quantum theory, various matters and different phases, etc. Fourth, the duality exists probably only under interactions. Some nonlinear quantum equations and wave equations have soliton solutions, which correspond to particles. This is mathematical description of wave-particle duality. Fifth, based on the soliton-chaos double solutions of nonlinear equations, we propose the field-particle duality, and field (wave)-quantum-chaos ternary. Sixth, the irreversibility and statistics are discussed, and we derive operators of entropy, etc. New measuring technology and some developed theories already challenge the uncertainty principle and duality, which may not be held under certain conditions, and must be modified and developed.
The Heisenberg uncertainty principle is one of the characteristics of quantum mechanics. With the vigorous development of quantum information theory, uncertain relations have gradually played an important role in it. In particular, in order to solved the shortcomings of the concept in the initial formulation of the uncertainty principle, we brought entropy into the uncertainty relation, after that, the entropic uncertainty relation has exploited the advantages to the full in various applications. As we all know the entropic uncertainty relation has became the core element of the security analysis of almost all quantum cryptographic protocols. This review mainly introduces development history and latest progress of uncertain relations. After Heisenberg's argument that incompatible measurement results are impossible to predict, many scholars, inspired by this viewpoint, have made further relevant investigations. They combined the quantum correlation between the observable object and its environment, and carried out various generalizations of the uncertainty relation to obtain more general formulas. In addition, it also focuses on the entropy uncertainty relationship and quantum-memory-assisted entropic uncertainty relation, and the dynamic characteristics of uncertainty in some physical systems. Finally, various applications of the entropy uncertainty relationship in the field of quantum information are discussed, from randomnesss to wave-particle duality to quantum key distribution.
Wave-particle duality is often considered as the modern answer to the problem of the nature of light after more than 2000 years of questioning. It is also the answer given by quantum physics concerning the nature of matter particles and any other radiations. The main objective of this work is to analyze the relations that are existing between this concept of wave-particle duality, the uncertainty principle and the concepts of phase space and microstates considered in statistical mechanics. It is mainly highlighted that while the concepts of phase space and microstates were already introduced in classical physics before the discovery of the wave-particle duality, a correct understanding of them cannot be achieved without the use of the concept of quantum phase space and phase space representation of quantum mechanics which are directly related to the uncertainty principle. The possibility of using these concepts of quantum phase space and phase space representations of quantum mechanics to help in a deeper description of the wave-particle duality and in the study of some current issues related to foundational problems of quantum mechanics like quantum decoherence and the measurement problem is also discussed.
First, we point out that the present applied superposition principle is linear, it must be developed into a generality. Next, the linear operators and equations should be developed nonlinearly. They will include nonlinear Klein-Gordon equation and Dirac equations, and corresponding Heisenberg equation. The quantum commutation and anticommutation should be developed. This theory may include the renormalization, which is the correction of Feynman rules of curved closed loops. We think the interaction equations are nonlinear. Many theories, models and phenomena are all nonlinear, for instance, soliton, nonabelian gauge field, and the bag model, etc. The superluminal entangled state, which relates the nonlocal quantum teleportation and nonlinearity, should be a new fifth interaction. Moreover, the NL effects exist possibly for four interactions, for single particle, for high energy, and for small space-time, etc. The relations among NL theory and electroweak unified theory, and QCD, and CP nonconservation, etc., are expounded. Finally, some known and possible tests are discussed. The NL theory relates the possible decrease of entropy in isolated system.
Bohr’s complementarity principle was introduced as a qualitative statement about quantum systems, which have properties that are equally real, but mutually exclusive. This principle, together with the uncertainty principle, is at the origin of Quantum Mechanics (QM), following the development of the theory since then. Another intriguing aspect of QM is entanglement, which is a type of correlation that is only possible within the mathematical formalism of QM. Its central importance in the Quantum foundations, as well as its important role in the fields of quantum information and quantum computation, has made the theory of entanglement achieve great progress in recent decades. In this dissertation we discuss the aspects and relations among complementarity relations, uncertainty, and entanglement. Such aspects are also investigated in relativistic scenarios, since the interest on how entanglement behaves in relativistic scenarios has grown more and more. First, by exploring the properties of the density operator, we obtain complementarity relations, using coherence measures well known in the literature. Next, we show that it is possible to obtain complete complementarity relations, i.e., relations that also take into account entanglement. For this, we explore the purity of a multipartite quantum system. In addition, we discuss the relation between complementarity and uncertainty of an observable, since it is possible to decompose uncertainty into its classical and quantum parts. We also obtain complementarity relations for uncertainty. Finally, we study the complete complementary relations and its Lorentz invariance. Although it is known that entanglement entropy does not remain invariant under Lorentz transformations, and neither does the measures that quantifies complementarity, we show that theses measures when taken together, in a complete complementarity relation, are Lorentz invariant.
This special issue is composed mainly of extended versions of talks and papers presented at the Seventh International Conference on Squeezed States and Uncertainty Relations held in Boston, Massachusetts, USA, on 4-8 June 2001. The conference was hosted by Boston University. The first meeting of this kind took place in College Park, Maryland, USA in 1991 as a small workshop. The second and third meetings were held in Moscow, Russia and Baltimore, USA in 1992 and 1993, respectively. The fourth meeting took place in Taiyuan, Shanxi Province, China in 1995 as an international conference sponsored by the International Union of Pure and Applied Physics. The fifth and sixth meetings were held in Balatonfüred, Hungary and Naples, Italy in 1997 and 1999, respectively. The eighth conference will take place in Puebla, Mexico in 2003. The contributions to this special issue address, in the first instance, the foundations of quantum mechanics and quantum optics, and several modern topics which have been intensively developed during the last decade. Quantum fluctuations, squeezing phenomena and different kinds of nonclassical states of light represent some of the most intensively discussed subjects. In particular, several papers deal with the properties of different kinds of `Schrödinger cat states' modelled in the simplest case by the even and odd coherent states. This paradigm exhibits in the most distinct form such cornerstone properties of quantum mechanics as the superposition principle, which manifests itself in quantum interference and entanglement. The quantum entanglement phenomenon has been studied intensively for the past few years due to its importance, for example, in quantum computer projects. In spite of the fact that this concept was known from the beginning of quantum mechanics, its properties, for example, the most appropriate entanglement measures, are still areas of discussion. Another related aspect of quantum nature is teleportation, which is also a subject of intensive study. The phenomena of dissipations and decoherence of the initial pure states are also very important, because fast decoherence can destroy all the advantages of quantum processes in quantum computing, teleportation and image processing. Due to these aspects, papers concerning various methods of controlling the decoherence, for example, by means of using different kinds of nonlinearities and deformations, are also presented here. From the very beginning of quantum mechanics, the uncertainty relations were basic inequalities distinguishing the classical and quantum worlds. Developing a deeper understanding of the well-known uncertainty relations and striving to find new ones (connected with purity parameters of states, entropy, etc.) is an important aspect of modern research. Recently developed tomographic methods of measuring quantum states (in terms of the Wigner function or other phase-space characteristics) have resulted in the formulation of quantum mechanics in which the standard probability density describes the quantum states in an alternative way to the density matrix or wave function. Some papers devoted to tomographic methods of measuring quantum states and studying different sorts of uncertainty relations are presented here. Group-theoretical methods of studying quantum-mechanical and quantum-optical phenomena, including the group-representation description of light polarization, also attract ongoing interest in the conference series discussions. We hope that this special issue will help readers to obtain an overview of the current trends in modern quantum optics, both experimental and theoretical. Young S Kim University of Maryland, College Park, USA Margarita A Man'ko P.N. Lebedev Physical Institute, Moscow, Russia Alexander Sergienko Boston University, MA, USA Guest Editors
Various dualities are summarized. Based on the universal wave-particle duality, along an opposite direction of the developed quantum mechanics, we use a method where the wave quantities frequency and wave length are replaced on various mechanical equations, and may be derive some new results. It is called the mechanical wave theory. From this we derive new operators which represent more physical quantities. Further, we propose some nonlinear equations and their solutions, which may be probably applied to quantum theory.