Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations

Newton-Leibniz integration rule only applies to commuting functions of continuum variables, while operators made of Dirac’s symbols (ket versus bra, e.g., |q〉〈q| of continuous parameter q) in quantum mechanics are usually not commutative. Therefore, integrations over the operators of type |〉〈| cannot be directly performed by Newton-Leibniz rule. We invented an innovative technique of integration within an ordered product (IWOP) of operators that made the integration of non-commutative operators possible. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics, and further reveals the beauty and elegance of Dirac’s symbolic method and transformation theory. Various applications of the IWOP technique, including constructing the entangled state representations and their applications, are presented.     

Stationary entanglement between macroscopic mechanical oscillators

We show that the optomechanical coupling between an optical cavity mode and two movable cavity mirrors is able to entangle two different macroscopic oscillation modes of the mirrors. This continuous variable entanglement is maintained by the light bouncing between the mirrors and is robust against thermal noise. In fact, it could be experimentally demonstrated using present technology. Received 2 September 2002 / Received in final form 10 October 2002 Published online 7 January 2003     

Rotationally Invariant Proof of Bell’s Theorem Without Inequalities for 4k-Dimensional (k=1,2,...) Systems

We present rotationally invariant proof of Bell‘s theorem without inequalities for spin-(2N 1)/2system (N = 2k-1,k = 1, 2, 3, 4...) with only two particles, which is a generalization of Cabello‘s study [Phys. Rev. A 67 (2003)032107].     

Teleportation for an Ionic Entangled Internal State by Entanglement Swapping

We present an effective scheme to teleport an unknown ionic entangled internal state via trapped ions without joint Bell-state measurement. In the constructed quantum channel process, we adopt entanglement swapping to avoid decrease of entanglement during the distribution of particles. Thus our scheme provides new prospects for quantum teleportation over longer distance. The distinct advantages of our scheme are that our scheme is insensitive to heating of vibrational mode and can be generalized to teleport an N-ion electronic entangled GHZ class state. Furthermore, in our scheme the success probability can reach 1.     

Generation of entanglement via atomic coherence in a two-mode three-level cascade atomic system

The generation of continuous variable entanglement via atomic coherence in a two-mode three-level cascade atomic system is discussed according to the entanglement criterion proposed by Duan et al. [Phys. Rev. Lett. 84, 2722 (2000)]. Atomic coherence between the top and bottom levels is induced with two photons of a strong external pump field. It shows that entanglement for the two-mode field in the cavity can be generated under certain conditions. Moreover, by means of the input-output theory, we show that the two-mode entanglement could also be approached at the output.     

Effects of phase decoherence on the entanglement of a two-qubit anisotropic Heisenberg XYZ chain with an in-plane magnetic field

We investigate the phase decoherence effects on the entanglement of a two-qubit anisotropic Heisenberg model with a nonuniform magnetic field in the x–z-plane. As a measure of the entanglement, the concurrence of the system is calculated. It is shown that when the magnetic field is along the z-axis, the nonuniform and uniform components of the field have no influence on the entanglement for the cases of and , respectively. But when the magnetic field is not along the z-axis, both the uniform and the nonuniform components of the field will introduce the decoherence effects. It is found that the effects of the Heisenberg chain's anisotropy in the Z-direction on the entanglement are dependent on the direction of the field. Moreover, the larger the initial concurrence is, the higher value it will exhibit during the time evolution of the system for a proper set of the parameters ν, Δ, θ, γ , B and b.     

Qubit disentanglement in local squeezed reservoirs

We consider the influence of the local squeezed vacuum fields on two initially entangled two-qubit system. By considering the upper bound of entanglement under time evolution, we find that the decay of the quantum entanglement shows different behavior for different time scales (t?max{(2βA)⁻¹,(2βB)⁻¹}$t?\max \{{(2{\beta }_A)}^{\mathrm{-1}},{(2{\beta }_B)}^{\mathrm{-1}}\}$ and t?min{(2βA)⁻¹,(2βB)⁻¹}$t?\min \{{(2{\beta }_A)}^{\mathrm{-1}},{(2{\beta }_B)}^{\mathrm{-1}}\}$). The relative phase of the squeezing environment can also affect the entanglement dynamics profoundly.     

The bound of entanglement of superpositions with more than two components

A bipartite quantum state (for two systems in any dimensions) can be decomposed as a superposition of many components. For a superposition of more than two components we prove that there is a bound of the entanglement of the superposition state which can be expressed according to entanglements of its component states. Especially, if the component states are mutually bi-orthogonal, the entanglement of the superposition state can be exactly given in terms of the entanglements of the states being superposed.     

Analysis of a recent experiment seemingly favouring local realism against quantum mechanics

A recent experiment by Brida et al. [Eur. Phys. J. D 44, 547 (2007)] is analyzed with the conclusion that the results disagree with standard quantum predictions but fit a simple local hidden variables model. New experiments are proposed which might throw new light on the anomaly.     

Boundary Quantum Entanglement of the XXZ Spin Chain with Boundary Impurities

The boundary quantum entanglement for the s = 1/2 X X Z spin chain with boundary impurities is studied via the density matrix renormalization group （DMRG） method. It is shown that the entanglement entropy of the boundary bond （the impurity and the chain spin next to it） behaves differently in different phases. The relationship between the singular points of the boundary entropy and boundary quantum critical points is discussed.