Entanglement under restricted operations: Analogy to mixed-state entanglement

We show that the classification of bipartite pure entangled states when local quantum operations are restricted yields a structure that is analogous in many respects to that of mixed-state entanglement. Specifically, we develop this analogy by restricting operations through local superselection rules, and show that such exotic phenomena as bound entanglement and activation arise using pure states in this setting. This analogy aids in resolving several conceptual puzzles in the study of entanglement under restricted operations. In particular, we demonstrate that several types of quantum optical states that possess confusing entanglement properties are analogous to bound entangled states. Also, the classification of pure-state entanglement under restricted operations can be much simpler than for mixed-state entanglement. For instance, in the case of local Abelian superselection rules all questions concerning distillability can be resolved.

Mixed-state entanglement in the light of pure-state entanglement constrained by superselection rules

We show that the classification of bi-partite pure entangled states when local quantum operations are restricted, e.g., constrained by local superselection rules, yields a structure that is analogous in many respects to that of mixed-state entanglement, including such exotic phenomena as bound entanglement and activation. This analogy aids in resolving several conceptual puzzles in the study of entanglement under restricted operations. Specifically, we demonstrate that several types of quantum optical states that possess confusing entanglement properties are analogous to bound entangled states. Also, the classification of pure-state entanglement under restricted operations can be much simpler than for mixed state entanglement. For instance, in the case of local Abelian superselection rules all questions concerning distillability can be resolved.

Entanglement of two-mode Bose-Einstein condensates

We investigate the entanglement characteristics of two general bimodal Bose-Einstein condensates-a pair of tunnel-coupled Bose-Einstein condensates and the atom-molecule Bose-Einstein condensate. We argue that the entanglement is only physically meaningful if the system is viewed as a bipartite system, where the subsystems are the two modes. The indistinguishibility of the particles in the condensate means that the atomic constituents are physically inaccessible and, thus, the degree of entanglement between individual particles, unlike the entanglement between the modes, is not experimentally relevant so long as the particles remain in the condensed state. We calculate the entanglement between the two modes for the exact ground state of the two bimodal condensates and consider the dynamics of the entanglement in the tunnel-coupled case.

Entanglement and bifurcations in Jahn-Teller models

We compare and contrast the entanglement in the ground state of two Jahn-Teller models. The Exbeta system models the coupling of a two-level electronic system, or qubit, to a single-oscillator mode, while the Exepsilon models the qubit coupled to two independent, degenerate oscillator modes. In the absence of a transverse magnetic field applied to the qubit, both systems exhibit a degenerate ground state. Whereas there always exists a completely separable ground state in the Exbeta system, the ground states of the Exepsilon model always exhibit entanglement. For the Exbeta case we aim to clarify results from previous work, alluding to a link between the ground-state entanglement characteristics and a bifurcation of a fixed point in the classical analog. In the Exepsilon case we make use of an ansatz for the ground state. We compare this ansatz to exact numerical calculations and use it to investigate how the entanglement is shared between the three system degrees of freedom.

Stationary two-atom entanglement induced by nonclassical two-photon correlations

A system of two two-level atoms interacting with a squeezed vacuum field can exhibit stationary entanglement associated with nonclassical two-photon correlations characteristic of the squeezed vacuum field. The amount of entanglement present in the system is quantified by the well known measure of entanglement called concurrence. We find analytical formulae describing the concurrence for two identical and nonidentical atoms and show that it is possible to obtain a large degree of steady-state entanglement in the system. Necessary conditions for the entanglement are nonclassical two-photon correlations and nonzero collective decay. It is shown that nonidentical atoms are a better source of stationary entanglement than identical atoms. We discuss the optimal physical conditions for creating entanglement in the system; in particular, it is shown that there is an optimal and rather small value of the mean photon number required for creating entanglement.

Dynamical creation of entanglement by homodyne-mediated feedback

For two two-level atoms coupled to a single Bosonic mode that is driven and heavily damped, the steady state can be entangled by resonantly driving the system [S. Schneider and G. J. Milburn, Phys. Rev. A 65, 042107 (2002)]. We present a scheme to significantly increase the steady-state entanglement by using homodyne-mediated feedback, in which the Bosonic mode is that of an electromagnetic cavity, the output of which is measured and the resulting homodyne photocurrent is used to modulate the field driving the qubits. Such feedback can increase the nonlinear response to both the decoherence process of the two-qubit system and the coherent evolution of individual qubits. We present the properties of the entangled states using the SO(3) Q function.

Entanglement measure for rank-2 mixed states

We provide an easily computable formula for a bipartite mixed-state entanglement measure. Our formula can be applied to readily calculate the entanglement for any rank-2 mixed state of a bipartite system. We use this formula to provide a tight upper bound for the entanglement of formation for rank-2 states of a qubit and a qudit. We also outline situations where our formula could be applied to study the entanglement properties of complex quantum systems.

Quantum computation by communication

We present here a new approach to scalable quantum computing - a 'qubus computer' - which realizes qubit measurement and quantum gates through interacting qubits with a quantum communication bus mode. The qubits could be 'static' matter qubits or 'flying' optical qubits, but the scheme we focus on here is particularly suited to matter qubits. There is no requirement for direct interaction between the qubits. Universal two-qubit quantum gates may be effected by schemes which involve measurement of the bus mode, or by schemes where the bus disentangles automatically and no measurement is needed. In effect, the approach integrates together qubit degrees of freedom for computation with quantum continuous variables for communication and interaction.

Multipartite entanglement and quantum state exchange

We investigate multipartite entanglement in relation to the process of quantum state exchange. In particular, we consider such entanglement for a certain pure state involving two groups of N trapped atoms. The state, which can be produced via quantum state exchange, is analogous to the steady-state intracavity state of the subthreshold optical nondegenerate parametric amplifier. We show that, first, it possesses some 2N-way entanglement. Second, we place a lower bound on the amount of such entanglement in the state using a measure called the entanglement of minimum bipartite entropy.

Entanglement induced by spontaneous emission in spatially extended two-atom systems

The role of the collective antisymmetric state in entanglement creation by spontaneous emission in a system of two non-overlapping two-level atoms has been investigated. Populations of the collective atomic states and the Wootters entanglement measure (concurrence) for two sets of initial atomic conditions are calculated and illustrated graphically. Calculations include the dipole-dipole interaction and a spatial separation between the atoms that the antisymmetric state of the system is included throughout even for small interatomic separations. It is shown that spontaneous emission can lead to a transient entanglement between the atoms even if the atoms were prepared initially in an unentangled state. It is found that the ability of spontaneous emission to create transient entanglement relies on the absence of population in the collective symmetric state of the system. For the initial state of only one atom excited, entanglement builds up rapidly in time and reaches a maximum for parameter values corresponding roughly to zero population in the symmetric state. On the other hand, for the initial condition of both atoms excited, the atoms remain unentangled until the symmetric state is depopulated. A simple physical interpretation of these results is given in terms of the diagonal states of the density matrix of the system. We also study entanglement creation in a system of two non-identical atoms of different transition frequencies. It is found that the entanglement between the atoms can be enhanced compared to that for identical atoms, and can decay with two different time scales resulting from the coherent transfer of the population from the symmetric to the antisymmetric state. In addition, it was found that a decaying initial entanglement between the atoms can display a revival behaviour.

Entanglement and spin squeezing in the two-atom Dicke model

We analyse the relation between the entanglement and spin-squeezing parameter in the two-atom Dicke model and identify the source of the discrepancy recently reported by Banerjee (2001 Preprint quant-ph/0110032) and Zhou et al (2002 J. Opt. B. Quantum Semiclass. Opt. 4 425), namely that one can observe entanglement without spin squeezing. Our calculations demonstrate that there are two criteria for entanglement, one associated with the two-photon coherences that create two-photon entangled states, and the other associated with populations of the collective states. We find that the spin-squeezing parameter correctly predicts entanglement in the two-atom Dicke system only if it is associated with two-photon entangled states, but fails to predict entanglement when it is associated with the entangled symmetric state. This explicitly identifies the source of the discrepancy and explains why the system can be entangled without spin squeezing. We illustrate these findings with three examples of the interaction of the system with thermal, classical squeezed vacuum, and quantum squeezed vacuum fields.

Entanglement of two atoms

We discuss the problem of creation of entangled states in a system of two two-level atoms which are separated by an arbitrary distance r(12) and interact with each other via the dipole-dipole interaction and both are driven by a laser field. The entangled antisymmetric state of the system is included throughout, even for small inter-atomic separations. Different mechanisms leading to effective transfer of population to the antisymmetric state are identified. The steady-state values of concurrence which is a measure of entanglement are calculated showing that perfect entanglement can be reached in case of two non-identical atoms.

Experimental characterization of continuous-variable entanglement

We present an experimental analysis of quadrature entanglement produced from a pair of amplitude squeezed beams. The correlation matrix of the state is characterized within a set of reasonable assumptions, and the strength of the entanglement is gauged using measures of the degree of inseparability and the degree of Einstein-Podolsky-Rosen (EPR) paradox. We introduce controlled decoherence in the form of optical loss to the entangled state, and demonstrate qualitative differences in the response of the degrees of inseparability and EPR paradox to this loss. The entanglement is represented on a photon number diagram that provides an intuitive and physically relevant description of the state. We calculate efficacy contours for several quantum information protocols on this diagram, and use them to predict the effectiveness of our entanglement in those protocols.

The effect of finite bandwidth squeezed light on entanglement creation in the Dicke model

We analyse the relation between local two-atom and total multi-atom entanglements in the Dicke system composed of a large number of atoms. We use concurrence as a measure of entanglement between two atoms in the multi-atom system, and the spin squeezing parameter as a measure of entanglement in the whole n-atom system. In addition, the influence of the squeezing phase and bandwidth on entanglement in the steady-state Dicke system is discussed. It is shown that the introduction of a squeezed field leads to a significant enhancement of entanglement between two atoms, and the entanglement increases with increasing degree of squeezing and bandwidth of the incident squeezed field. In the presence of a coherent field the entanglement exhibits a strong dependence on the relative phase between the squeezed and coherent fields, that can jump quite rapidly from unentangled to strongly entangled values when the phase changes from zero to pi. We find that the jump of the degree of entanglement is due to a flip of the spin squeezing from one quadrature component of the atomic spin to the other component when the phase changes from zero to pi. We also analyse the dependence of the entanglement on the number of atoms and find that, despite the reduction in the degree of entanglement between two atoms, a large entanglement is present in the whole n-atom system and the degree of entanglement increases as the number of atoms increases.