Violations of Bell's Inequality in Synchronized Hyperchaos

Motivated by a parallel between quantum cryptography and chaos synchronization cryptography, we construct a Bell's inequality for a pair of synchronously coupled variable-order Generalized Rossler Systems, with arbitrarily binarized final states. In the infinite-order limit, although dynamical parameters cannot be extracted from the coupling signal in finite time, the inequality is violated, as with entangled quantum states. The violations are weaker than in quantum theory, vanishing as the differences between corresponding parameters of the coupled systems become small. The fact that Bell's inequality can be violated for a pair of classical systems that are not discernibly connected supports the possibility of a realist interpretation of quantum mechanics.     

On Algebraic Integrability of the Deformed Elliptic Calogero-Moser Problem

Abstract

We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the ? ⁴ type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or irregular quantum states. As a first (typical) example we consider a single electron which is strongly coupled with phonons on a 1D chain of atoms — the (Rashba)–Holstein polaron model. In the adiabatic approximation this system is conventionally described by the DNSE. Another relevant example is that of superconducting states in layered superconductors described by the same DNSE. Amongst many other applications the typical example for a classical lattice is a system of coupled nonlinear oscillators. We present the exact energy spectrum of this model in the strong coupling limit and the corresponding wave function. Using this as a starting point we go on to calculate the wave function for moderate coupling and find that the energy eigenvalue of these structures of the wave function is in exquisite agreement with the exact strong coupling result. This procedure allows us to obtain (numerically) exact solutions of the DNSE directly. When applied to our typical example we find that the wave function of an electron on a deformable lattice (and other quantum or classical discrete systems) may exhibit incommensurate and irregular structures. These states are analogous to the periodic, quasiperiodic and chaotic structures found in classical chaotic dynamics.     

Hybrid stable-chaotic states in coupled quantum well stadia

We use tunnel current spectroscopy to investigate the quantum states of two GaAs quantum wells coupled by a low (100 meV) (AlGa)As tunnel barrier. A high tilted magnetic field is used to generate strongly chaotic electron motion in the two wells which act as coupled chaotic ‘stadia'. The effect of the tunnel barrier on the dynamics of the system depends on the magnitude of the applied bias voltage V. For V375 mV, the central potential barrier acts as a perturbation which modifies the trajectories of selected periodic orbits in the quantum well. Scattering off the central barrier also generates new periodic orbits involving multiple collisions on all three barriers. These orbits ‘scar' distinct sets of eigenstates which generate periodic resonant peaks in the current–voltage characteristics of the device. When the device is biased such that the injected electrons just surmount the central barrier, our calculations reveal novel hybrid scarred states with both stable and chaotic characteristics.     

Activating the Violation of Steering Inequality for Two‐Qubit X States

The phenomenon of superactivation of quantum steering, that is, the fact that by combining several copies of the unsteerable quantum state one can get a steerable quantum state, has recently attracted much attention. This property is currently only known for isotropic states, and whether there exists an example other than isotropic states is so far an open question. It is of great importance to learn the activation property of non‐isotropic states. In this paper, it is shown that the violation of the local uncertainty relations (LUR) steering inequality can be activated for two‐qubit X type entangled mixed states. Although the activation phenomenon proven in this paper cannot guarantee the superactivation of steering for two‐qubit X states, it does provide clues for this goal. In addition, previous results indicate that two copies are enough for the superactivation of steering, while the price to pay is a high local dimension, and it is unclear whether superactivation is possible with few copies in low‐dimensional systems such as two‐qubit systems. By providing examples of activation of LUR steering inequality with two copies in a two‐qubit system, the work may open a potential way to the solution of this question.     

The characteristics of nonlinear chaotic dynamics in quantum cellular neural networks

With the polarization of quantum-dot cell and quantum phase serving as state variables, this paper does both theoretical analysis and simulation for the complex nonlinear dynamical behaviour of a three-cell-coupled Quantum Cellular Neural Network （QCNN）, including equilibrium points, bifurcation and chaotic behaviour. Different phenomena, such as quasi-periodic, chaotic and hyper-chaotic states as well as bifurcations are revealed. The system＇s bifurcation and chaotic behaviour under the influence of the different coupling parameters are analysed. And it finds that the unbalanced cells coupled QCNN is easy to cause chaotic oscillation and the system response enters into chaotic state from quasi-periodic state by quasi-period bifurcation; however, the balanced cells coupled QCNN also can be chaotic when coupling parameters is in some region. Additionally, both the unbalanced and balanced cells coupled QCNNs can possess hyper-chaotic behaviour. It provides valuable information about QCNNs for future application in high-parallel signal processing and novel ultra-small chaotic generators.     

Electron entanglement detected by quantum spin Hall systems

We propose a promising electron entanglement detector consisting of two quantum spin Hall systems weakly coupled to a superconductor. The detection of electron spins along various polarization directions, which is a prerequisite for testing Bell's inequality on solid state spins, can be achieved in an all-electrical-controlled manner utilizing the helical edge states. It is found that the violation of Bell's inequality exists in a large range of the tunneling parameters, which can be realized in mercury telluride quantum wells.     

Separability of Quantum States vs. Original Bell (1964) Inequalities

All separable states satisfy all Bell-type inequalities, which involve as their assumption only existence of local realistic (local hidden variable) models of the correlations of spatially separated systems, observed by two or more observers making independent decisions on what to measure (free will). The recent observation by Loubenets, that some separable states do not satisfy the original Bell inequality (1964) has no consequences whatsoever for the studies of the relation of separability with local realism. The original Bell inequality was derived using an additional assumption that the local results for a certain pair of local settings reveal perfect Einstein–Podolsky–Rosen (EPR) correlations. Therefore violation of this inequality by some quantum predictions implies that either (i) the predictions do not allow a local realistic model, or (ii) the predictions do not have the required EPR correlations, or finally both (i) and (ii).     

Quantum qualitative dynamics

We describe our work on qualitative methods for visualizing the quantum eigenstates of systems with nonlinear classical dynamics. For two-degree-of-freedom systems, our approach is based on the use of generalized coherent states, and allows systems with nonoscillator kinematics to be investigated. The general approach is illustrated with two examples involving vibration-rotation interaction in polyatomic molecules. We apply the coherent states of the Lie groupH ₄SU(2) to define quantum surfaces of section for a model involving centrifugal coupling of a harmonic bend with molecular rotation, andSU(2)SU(2) coherent states to study two harmonic normal modes coupled to overall molecular rotation through coriolis interaction. In both systems, quantum states are visualized on the rotational surface of section and compared with the corresponding classical phase space structure. Striking classical-quantum correspondence is observed. We then describe recent results on the quantum states of (N 3)-dimensional systems of coupled nonlinear oscillators, which reveal a quantum delocalization that is reminiscent of classical Arnold diffusion.     

Transport through artificial single-molecule magnets: Spin-pair state sequential tunneling and Kondo effects

The transport properties of an artificial single-molecule magnet based on a CdTe quantum dot doped with a single Mn+2 ion(S=5/2) are investigated by the non-equilibrium Green function method.We consider a minimal model where the Mn-hole exchange coupling is strongly anisotropic so that spin-flip is suppressed and the impurity spin S and a hole spin s entering the quantum dot are coupled into spin pair states with(2S+1) sublevels.In the sequential tunneling regime,the differential conductance exhibits(2S+1) possible peaks,corresponding to resonance tunneling via(2S+1) sublevels.At low temperature,Kondo physics dominates transport and(2S+1) Kondo peaks occur in the local density of states and conductance.These peaks originate from the spin-singlet state formed by the holes in the leads and on the dot via higher-order processes and are related to the parallel and antiparallel spin pair states.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

The equivalence of Bell's inequality and the Nash inequality in a quantum game-theoretic setting

The interaction of competing agents is described by classical game theory. It is now well known that this can be extended to the quantum domain, where agents obey the rules of quantum mechanics. This is of emerging interest for exploring quantum foundations, quantum protocols, quantum auctions, quantum cryptography, and the dynamics of quantum cryptocurrency, for example. In this paper, we investigate two-player games in which a strategy pair can exist as a Nash equilibrium when the games obey the rules of quantum mechanics. Using a generalized Einstein–Podolsky–Rosen (EPR) setting for two-player quantum games, and considering a particular strategy pair, we identify sets of games for which the pair can exist as a Nash equilibrium only when Bell's inequality is violated. We thus determine specific games for which the Nash inequality becomes equivalent to Bell's inequality for the considered strategy pair.     

Tripartite states Bell-nonlocality sudden death in a spin[1mm] environment with multisite interaction

Tis paper demonstrates that multipartite Bell-inequality violations can be fully destroyed in a finite time in three-qubit states coupled to a general XY spin-chain with a three-site interaction environment. The Mermin—Ardehali—Belinksii—Klyshko inequality is used to detect the degree of nonlocality, as measured by the extent of their violations. The effects of system-environment couplings, the size of degrees of freedom of the environment and the strength of the three-site interaction on the Bell-inequality violations are given. The results indicate that the Bell-inequality violations of the tripartite states will be completely destroyed by decoherence under certain conditions for the GHZ state. The decoherence-free subspaces of our model are identified and the entanglement of quantum states is also discussed.     

The non-Markovian quantum behavior of open systems

It is shown that the exact dynamics of a composite quantum system can be represented through a pair of product states which evolve according to a Markovian random jump process. This representation is used to design a general Monte Carlo wave function method that enables the stochastic treatment of the full non-Markovian behavior of open quantum systems. Numerical simulations are carried out which demonstrate that the method is applicable to open systems strongly coupled to a bosonic reservoir, as well as to the interaction with a spin bath. Full details of the simulation algorithms are given, together with an investigation of the dynamics of fluctuations. Several potential generalizations of the method are outlined.Received: 29 October 2003, Published online: 10 February 2004PACS: 03.65.Yz Decoherence; open systems; quantum statistical methods - 02.70.Ss Quantum Monte Carlo methods - 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)     

Randomized Entangled Mixed States from Phase States

We construct randomized entangled mixed states by using the formalism of phase states for d-dimensional systems (qudits). The randomized entangled mixed states are a special kind of mixed states that exhibit genuine multipartite correlation. Such states are obtained by the application of randomized entangling operators to an arbitrary pair of qudits of a multiqudit system. The study of the entanglement of randomized mixed states is of great importance in quantum computation since any experimental implementation of entangled states in a realistic environment can be made by imperfect entangling gates. We give a brief review of some necessary background about unitary phase operators and phase states of a multi-qudit system. Evolved density matrices arise when qudits of the multi-qudit system interact via a Hamiltonian of Heisenberg type. The randomized entangled states associated with evolved density matrices are derived via the action of an entangling operator on a pair of two qudits {i, j} of the multi-qudit system with some probability p. The randomized entangled mixed states for bipartite, tripartite and multipartite systems are explicitly expressed and their Kraus decomposition properties are discussed.

     

Chaotic Entanglement: Entropy and Geometry

In chaotic entanglement, pairs of interacting classically-chaotic systems are induced into a state of mutual stabilization that can be maintained without external controls and that exhibits several properties consistent with quantum entanglement. In such a state, the chaotic behavior of each system is stabilized onto one of the system’s many unstable periodic orbits (generally located densely on the associated attractor), and the ensuing periodicity of each system is sustained by the symbolic dynamics of its partner system, and vice versa. Notably, chaotic entanglement is an entropy-reversing event: the entropy of each member of an entangled pair decreases to zero when each system collapses onto a given period orbit. In this paper, we discuss the role that entropy plays in chaotic entanglement. We also describe the geometry that arises when pairs of entangled chaotic systems organize into coherent structures that range in complexity from simple tripartite lattices to more involved patterns. We conclude with a discussion of future research directions.     

Experimental Investigation of the Robustness of a New Bell-Type Inequality of Triphoton GHZ States in Open Systems

Knowing the level of entanglement robustness against quantum bit loss or decoherence mechanisms is an important issue for any application of quantum information. Fidelity of states can be used to judge whether there is entanglement in multi-particle systems. It is well known that quantum channel security in QKD can be estimated by measuring the robustness of Bell-type inequality against noise. We experimentally investigate a new Bell-type inequality (NBTI) in the three-photon Greenberger–Horne–Zeilinger (GHZ) states with different levels of spin-flip noise. The results show that the fidelity and the degree of violation of the NBTI decrease monotonically with the increase of noise intensity. They also provide a method to judge whether there is entanglement in three-particle mixed states.     

Deterministic Entanglement Purification of the Greenberger-Horne-Zeilinger States in Quantum-Dot and Micro-cavity Coupled System

An improved purification of the triplet Greenberger-Horne-Zeilinger (GHZ) state is demonstrated for the electron-spin-entangled state in the quantum-dot (QD) and micro-cavity coupled systems. In order to distill the maximally entangled GHZ state efficiently, we designate a deterministic entanglement purification protocol (EPP) by using a pair of the triplet-electron-spin-entangled systems. It is based on the elegant parity-check operations performed in the cavity-spin-coupling system with the assistance of an ancillary single photon. With the current and feasible technology, the maximally entangled GHZ states can be achieved as much as flexible for the long-distance quantum communications since only single-photon detection and single-electron detection are required in practice.     

The search for a quantum KAM theorem

The complex mechanisms by which nonlinear classical conservative systems undergo a transition from quasiperiodic to chaotic behavior are now fairly well understood. This transition is associated with a breakdown of quasi-constants of motion (KAM surfaces). There is growing evidence that similar mechanisms may govern the behavior of quantum systems. While K-type mixing behavior has not yet been found, there does appear to be a transition associated with the destruction of a quantum quasi-constant of motion (quantum KAM states) which changes qualitatively the spectrum of quantum systems.     

Hopping of chaotic dynamics in a doubly resonant parametric oscillator

We study a doubly resonant optical parametric oscillator where the pump can feed two pairs of signal-idler modes. We assume the presence of gain at the pump frequency. We investigate the various oscillation states of interest, namely, when only the first pair oscillates with the other pair having null amplitudes and vice versa. We demonstrate the exchange of dynamics between the mode pairs when the relevant parameters of the cavity, namely, the phase mismatch factors or the decay rates switch because of fluctuations. The exchange of dynamics is shown to be independent of the nature of dynamics, i.e. independent of whether the motion isn-periodic or chaotic. We also investigate the case where both the pairs can exhibit chaotic dynamics though these states are difficult to realize because of fluctuations.