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We present a robust and fast laser cooling scheme suitable for trapped atoms and ions. Based on quantum interference, generated by a special laser configuration, it is able to rapidly cool the system such that the final phonon occupation vanishes to zeroth order in the Lamb-Dicke parameter in contrast to existing cooling schemes. Furthermore, it is robust under conditions of fluctuating laser intensity and frequency, thus making it a viable candidate for experimental applications., Comment: 5 pages, 3 figures

In recent work Hartmann et al [Phys. Rev. Lett. 102, 057202 (2009)] demonstrated that the classical simulation of the dynamics of open 1D quantum systems with matrix product algorithms can often be dramatically improved by performing time evolution in the Heisenberg picture. For a closed system this was exemplified by an exact matrix product operator solution of the time-evolved creation operator of a quadratic fermi chain with a matrix dimension of just two. In this work we show that this exact solution can be significantly generalized to include the case of an open quadratic fermi chain subjected to master equation evolution with Lindblad operators that are linear in the fermionic operators. Remarkably even in this open system the time-evolution of operators continues to be described by matrix product operators with the same fixed dimension as that required by the solution of a coherent quadratic fermi chain for all times. Through the use of matrix product algorithms the dynamical behaviour of operators in this non-equilibrium open quantum system can be computed with a cost that is linear in the system size. We present some simple numerical examples which highlight how useful this might be for the more detailed study of open system dynamics. Given that Heisenberg picture simulations have been demonstrated to offer significant accuracy improvements for other open systems that are not exactly solvable our work also provides further insight into how and why this advantage arises., Comment: 22 pages and 8 figures

Recent efforts have applied quantum tomography techniques to the calibration and characterization of complex quantum detectors using minimal assumptions. In this work we provide detail and insight concerning the formalism, the experimental and theoretical challenges and the scope of these tomographical tools. Our focus is on the detection of photons with avalanche photodiodes and photon number resolving detectors and our approach is to fully characterize the quantum operators describing these detectors with a minimal set of well specified assumptions. The formalism is completely general and can be applied to a wide range of detectors, Comment: 22 pages, 27 figures

Measurement is the only part of a general quantum system that has yet to be characterized experimentally in a complete manner. Detector tomography provides a procedure for doing just this; an arbitrary measurement device can be fully characterized, and thus calibrated, in a systematic way without access to its components or its design. The result is a reconstructed POVM containing the measurement operators associated with each measurement outcome. We consider two detectors, a single-photon detector and a photon-number counter, and propose an easily realized experimental apparatus to perform detector tomography on them. We also present a method of visualizing the resulting measurement operators., Comment: 9 pages, 4 figures

We investigate the entanglement between two separated segments in the vacuum state of a free 1D Klein-Gordon field, where explicit computations are performed in the continuum limit of the linear harmonic chain. We show that the entanglement, which we measure by the logarithmic negativity, is finite with no further need for renormalization. We find that the quantum correlations decay much faster than the classical correlations as in the critical limit long range entanglement decays exponentially for separations larger than the size of the segments. As the segments become closer to each other the entanglement diverges as a power law. The noncritical regime manifests richer behavior, as the entanglement depends both on the size of the segments and on their separation. In correspondence with the von Neumann entropy long-range entanglement also distinguishes critical from noncritical systems.

We consider the possibility of adding noise to a quantum circuit to make it efficiently simulatable classically. In previous works this approach has been used to derive upper bounds to fault tolerance thresholds - usually by identifying a privileged resource, such as an entangling gate or a non-Clifford operation, and then deriving the noise levels required to make it `unprivileged'. In this work we consider extensions of this approach where noise is added to Clifford gates too, and then `commuted' around until it concentrates on attacking the non-Clifford resource. While commuting noise around is not always straightforward, we find that easy instances can be identified in popular fault tolerance proposals, thereby enabling sharper upper bounds to be derived in these cases. For instance we find that if we take Knill's high threshold proposal together with the ability to prepare any possible state in the $XY$ plane of the Bloch sphere, then no more than 3.69% error-per-gate noise is sufficient to make it classical, and 13.71% of Knill's gamma noise model is sufficient. These bounds have been derived without noise being added to the decoding parts of the circuits. Introducing such noise in a toy example suggests that the present approach can be optimised further to yield tighter bounds.

We present a detailed study on the possibility of manipulating quantum information encoded in the "radial" modes of arrays of trapped ions (i.e., in the ions' oscillations orthogonal to the trap's main axis). In such systems, because of the tightness of transverse confinement, the radial modes pertaining to different ions can be addressed individually. In the first part of the paper we show that, if local control of the radial trapping frequencies is available, any linear optical and squeezing operation on the locally defined modes - on single as well as on many modes - can be reproduced by manipulating the frequencies. Then, we proceed to describe schemes apt to generate unprecedented degrees of bipartite and multipartite continuous variable entanglement under realistic noisy working conditions, and even restricting only to a global control of the trapping frequencies. Furthermore, we consider the transmission of the quantum information encoded in the radial modes along the array of ions, and show it to be possible to a remarkable degree of accuracy, for both finite-dimensional and continuous variable quantum states. Finally, as an application, we show that the states which can be generated in this setting allow for the violation of multipartite non-locality tests, by feasible displaced parity measurements. Such a demonstration would be a first test of quantum non-locality for "massive" degrees of freedom (i.e., for degrees of freedom describing the motion of massive particles)., Comment: 21 pages; this paper, presenting a far more extensive and detailed analysis, completely supersedes arXiv:0708.0851

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: The entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such "area laws" for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium we review the current status of area laws in these fields. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation, and disordered systems, non-equilibrium situations, classical correlation concepts, and topological entanglement entropies are discussed. A significant proportion of the article is devoted to the quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. We discuss matrix-product states, higher-dimensional analogues, and states from entanglement renormalization and conclude by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations., Comment: 28 pages, 2 figures, final version

The transfer of a quantum state between distant nodes in two-dimensional networks, is considered. The fidelity of state transfer is calculated as a function of the number of interactions in networks that are described by regular graphs. It is shown that perfect state transfer is achieved in a network of size N, whose structure is that of a N/2-cross polytope graph, if N is a multiple of 4. The result is reminiscent of the Babinet principle of classical optics. A quantum Babinet principle is derived, which allows for the identification of complementary graphs leading to the same fidelity of state transfer, in analogy with complementary screens providing identical diffraction patterns., Comment: 5 pages, 3 figures; Replaced with published version

We derive fidelity benchmarks for the quantum storage and teleportation of squeezed states of continuous variable systems, for input ensembles where the degree of squeezing $s$ is fixed, no information about its orientation in phase space is given, and the distribution of phase space displacements is a Gaussian. In the limit where the latter becomes flat, we prove analytically that the maximal classical achievable fidelity (which is 1/2 without squeezing, for $s=1$) is given by $\sqrt{s}/(1+s)$, vanishing when the degree of squeezing diverges. For mixed states, as well as for general distributions of displacements, we reduce the determination of the benchmarks to the solution of a finite-dimensional semidefinite program, which yields accurate, certifiable bounds thanks to a rigorous analysis of the truncation error. This approach may be easily adapted to more general ensembles of input states., Comment: 19 pages, 4figures

Transport phenomena are fundamental in Physics. They allow for information and energy to be exchanged between individual constituents of communication systems, networks or even biological entities. Environmental noise will generally hinder the efficiency of the transport process. However, and contrary to intuition, there are situations in classical systems where thermal fluctuations are actually instrumental in assisting transport phenomena. Here we show that, even at zero temperature, transport of excitations across dissipative quantum networks can be enhanced by local dephasing noise. We explain the underlying physical mechanisms behind this phenomenon, show that entanglement does not play a supportive role and propose possible experimental demonstrations in quantum optics. We argue that Nature may be routinely exploiting this effect and show that the transport of excitations in light harvesting molecules does benefit from such noise assisted processes. These results point towards the possibility for designing optimized structures for transport, for example in artificial nano-structures, assisted by noise.

Measurement connects the world of quantum phenomena to the world of classical events. It plays both a passive role, observing quantum systems, and an active one, preparing quantum states and controlling them. Surprisingly - in the light of the central status of measurement in quantum mechanics - there is no general recipe for designing a detector that measures a given observable. Compounding this, the characterization of existing detectors is typically based on partial calibrations or elaborate models. Thus, experimental specification (i.e. tomography) of a detector is of fundamental and practical importance. Here, we present the realization of quantum detector tomography: we identify the optimal positive-operator-valued measure describing the detector, with no ancillary assumptions. This result completes the triad, state, process, and detector tomography, required to fully specify an experiment. We characterize an avalanche photodiode and a photon number resolving detector capable of detecting up to eight photons. This creates a new set of tools for accurately detecting and preparing non-classical light., Comment: 6 pages, 4 figures,see video abstract at http://www.quantiki.org/video_abstracts/08072444

We introduce a class of variational states to describe quantum many-body systems. This class generalizes matrix product states which underly the density-matrix renormalization group approach by combining them with weighted graph states. States within this class may (i) possess arbitrarily long-ranged two-point correlations, (ii) exhibit an arbitrary degree of block entanglement entropy up to a volume law, (iii) may be taken translationally invariant, while at the same time (iv) local properties and two-point correlations can be computed efficiently. This new variational class of states can be thought of as being prepared from matrix product states, followed by commuting unitaries on arbitrary constituents, hence truly generalizing both matrix product and weighted graph states. We use this class of states to formulate a renormalization algorithm with graph enhancement (RAGE) and present numerical examples demonstrating that improvements over density-matrix renormalization group simulations can be achieved in the simulation of ground states and quantum algorithms. Further generalizations, e.g., to higher spatial dimensions, are outlined., Comment: 4 pages, 1 figure

A simple technique for robust generation of vibrational Fock states in a chain of trapped ions is proposed. The method is fast and easy to implement, since only a single chirped laser pulse, simultaneously addressing all of the ions, is required. Furthermore, because the approach uses collective adiabatic passage, significant fluctuations in the intensity or frequency of the laser pulse can be tolerated, and the technique performs well even on the border of the Lamb-Dicke regime. We also demonstrate how this technique may be extended in order to create non-classical superposition states of the ions' collective motion and Greenberger-Horne-Zeilinger states of their internal states. Because only a single laser pulse is required, heating effects arising under realistic experimental conditions are negligibly small., Comment: 9 pages, 7 figures. Discussion of performance outside Lamb-Dicke regime added. Some refs added

Since Hund's work on the ammonia molecule, the double well potential has formed a key paradigm in physics. Its importance is further underlined by the central role it plays in the Landau theory of phase transitions. Recently, the study of entanglement properties of many-body systems has added a new angle to the study of quantum phase transitions of discrete and continuous degrees of freedom, i.e., spin and harmonic chains. Here we show that control of the radial degree of freedom of trapped ion chains allows for the simulation of linear and non-linear Klein-Gordon fields on a lattice, in which the parameters of the lattice, the non-linearity and mass can be controlled at will. The system may be driven through a phase transition creating a double well potential between different configurations of the ion crystal. The dynamics of the system are controllable, local properties are measurable and tunnelling in the double well potential would be observable., Comment: 6 pages, 5 figures

In most studies of the capacity of quantum channels, it is assumed that the errors in each use of the channel are independent. However, recent work has begun to investigate the effects of memory or correlations in the error, and has led to suggestions that there can be interesting non-analytic behaviour in the capacity of such channels. In a previous paper we pursued this issue by connecting the study of channel capacities under correlated error to the study of critical behaviour in many-body physics. This connection enables the use of techniques from many-body physics to either completely solve or understand qualitatively a number of interesting models of correlated error with analogous behaviour to associated many-body systems. However, in order for this approach to work rigorously, there are a number of technical properties that need to be established for the lattice systems being considered. In this article we discuss these properties in detail, and establish them for some classes of many-body system., Comment: V1: 17 sides, 5 figures. See posting by M. Hastings 0710.3324 for related results on fermionic systems. V2: 19 sides, 5 figures, extensive changes to improve readability and to correct an imprecise argument. To appear in New J. Phys. Title change due to referee request

We propose and study methods for detecting the Unruh effect in a Bose-Einstein condensate. The Bogoliubov vacuum of a Bose-Einstein condensate is used here to simulate a scalar field-theory, and accelerated atom dots or optical lattices as means for detecting phonon radiation due to acceleration effects. We study Unruh's effect for linear acceleration and circular acceleration. In particular, we study the dispersive effects of the Bogoliubov spectrum on the ideal case of exact thermalization. Our results suggest that Unruh's acceleration radiation can be tested using current accessible experimental methods., Comment: 5 pages, 3 figures

In most studies of the channel capacity of quantum channels, it is assumed that the errors in each use of the channel are independent. However, recent work has begun to investigate the effects of memory or correlations in the error. This work has led to speculation that interesting non-analytic behaviour may occur in the capacity. Motivated by these observations, we connect the study of channel capacities under correlated error to the study of critical behaviour in many-body physics. This connection enables us the techniques of many-body physics to either completely solve or understand qualitatively a number of interesting models of correlated error. The models can display analogous behaviour to associated many-body systems, including `phase transitions'., Comment: V2: changes in presentation, some additional comments on generalisation. V3: In accordance with published version, most (but not all) details of proofs now included. A separate paper will shortly be submitted separately with all details and more results

We investigate the entanglement within a system undergoing a random, local process. We find that there is initially a phase of very fast generation and spread of entanglement. At the end of this phase the entanglement is typically maximal. In previous work we proved that the maximal entanglement is reached to a fixed arbitrary accuracy within $O(N^3)$ steps, where $N$ is the total number of qubits. Here we provide a detailed and more pedagogical proof. We demonstrate that one can use the so-called stabilizer gates to simulate this process efficiently on a classical computer. Furthermore, we discuss three ways of identifying the transition from the phase of rapid spread of entanglement to the stationary phase: (i) the time when saturation of the maximal entanglement is achieved, (ii) the cut-off moment, when the entanglement probability distribution is practically stationary, and (iii) the moment block entanglement scales exhibits volume scaling. We furthermore investigate the mixed state and multipartite setting. Numerically we find that classical and quantum correlations appear to behave similarly and that there is a well-behaved phase-space flow of entanglement properties towards an equilibrium, We describe how the emergence of typical entanglement can be used to create a much simpler tripartite entanglement description. The results form a bridge between certain abstract results concerning typical (also known as generic) entanglement relative to an unbiased distribution on pure states and the more physical picture of distributions emerging from random local interactions., Comment: 32 pages, 11 figures

We present a framework, compliant with the general canonical principle of statistical mechanics, to define measures on the set of pure Gaussian states of continuous variable systems. Within such a framework, we define two specific measures, referred to as `micro-canonical' and `canonical', and apply them to study systematically the statistical properties of the bipartite entanglement of n-mode pure Gaussian states (as quantified by the entropy of a subsystem). We rigorously prove the "concentration of measure" around a finite average, occurring for the entanglement in the thermodynamical limit in both the canonical and the micro-canonical approach. For finite n, we determine analytically the average and standard deviation of the entanglement (as quantified by the reduced purity) between one mode and all the other modes. Furthermore, we numerically investigate more general situations, clearly showing that the onset of the concentration of measure already occurs at relatively small n., Comment: 24 pages, 5 figures, IOP style; conclusions extended, minor layout adjustment

The geometric phase can act as a signature for critical regions of interacting spin chains in the limit where the corresponding circuit in parameter space is shrunk to a point and the number of spins is extended to infinity; for finite circuit radii or finite spin chain lengths, the geometric phase is always trivial (a multiple of 2pi). In this work, by contrast, two related signatures of criticality are proposed which obey finite-size scaling and which circumvent the need for assuming any unphysical limits. They are based on the notion of the Bargmann invariant whose phase may be regarded as a discretized version of Berry's phase. As circuits are considered which are composed of a discrete, finite set of vertices in parameter space, they are able to pass directly through a critical point, rather than having to circumnavigate it. The proposed mechanism is shown to provide a diagnostic tool for criticality in the case of a given non-solvable one-dimensional spin chain with nearest-neighbour interactions in the presence of an external magnetic field., Comment: 7 Figures

The entanglement entropy of a distinguished region of a quantum many-body system reflects the entanglement present in its pure ground state. In this work, we establish scaling laws for this entanglement for critical quasi-free fermionic and bosonic lattice systems, without resorting to numerical means. We consider the geometrical setting of D-dimensional half-spaces which allows us to exploit a connection to the one-dimensional case. Intriguingly, we find a difference in the scaling properties depending on whether the system is bosonic - where an area-law is first proven to hold - or fermionic, extending previous findings for cubic regions. For bosonic systems with nearest neighbor interaction we prove the conjectured area-law by computing the logarithmic negativity analytically. We identify a length scale associated with entanglement, different from the correlation length. For fermions we determine the logarithmic correction to the area-law, which depends on the topology of the Fermi surface. We find that Lifshitz quantum phase transitions are accompanied with a non-analyticity in the prefactor of the leading order term., Comment: 4 pages, 1 figure (essentially identical with published version)

The entanglement dynamics of arrays of qubits is analysed in the presence of some general sources of noise and disorder. In particular, we consider linear chains of Josephson qubits in experimentally realistic conditions. Electromagnetic and other (spin or boson) fluctuations due to the background circuitry and surrounding substrate, finite temperature in the external environment, and disorder in the initial preparation and the control parameters are embedded into our model. We show that the amount of disorder that is typically present in current experiments does not affect the entanglement dynamics significantly, while the presence of noise can have a drastic influence on the generation and propagation of entanglement. We examine under which circumstances the system exhibits steady-state entanglement for both short (N < 10) and long (N > 30) chains and show that, remarkably, there are parameter regimes where the steady-state entanglement is strictly non-monotonic as a function of the noise strength. We also present optimized schemes for entanglement verification and quantification based on simple correlation measurements that are experimentally more economic than state tomography., Comment: 17 pages, 9 figures (replaced with published version: updated references, minor changes and typos)

The verification and quantification of experimentally created entanglement by simple measurements, especially between distant particles, is an important basic task in quantum processing. When composite systems are subjected to local measurements the measurement data will exhibit correlations, whether these systems are classical or quantum. Therefore, the observation of correlations in the classical measurement record does not automatically imply the presence of quantum correlations in the system under investigation. In this work we explore the question of when correlations, or other measurement data, are sufficient to guarantee the existence of a certain amount of quantum correlations in the system and when additional information, such as the degree of purity of the system, is needed to do so. Various measurement settings are discussed, both numerically and analytically. Exact results and lower bounds on the least entanglement consistent with the observations are presented. The approach is suitable both for the bi-partite and the multi-partite setting., Comment: 10 pages and 4 figures, material and references added

A laser cooling scheme for trapped ions is presented which is based on the fast dynamical Stark shift gate, described in [Jonathan etal, PRA 62, 042307]. Since this cooling method does not contain an off resonant carrier transition, low final temperatures are achieved even in traveling wave light field. The proposed method may operate in either pulsed or continuous mode and is also suitable for ion traps using microwave addressing in strong magnetic field gradients., Comment: 4 pages 5 figures

We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process generates the average entanglement of the uniform (Haar) measure in at most $O(N^3)$ steps for $N$ qubits. This is despite an exponentially growing number of such gates being necessary for generating that measure fully on the state space. Numerics furthermore show a variation cut-off allowing one to associate a specific time with the achievement of the uniform measure entanglement distribution. Various extensions of this work are discussed. The results are relevant to entanglement theory and to protocols that assume generic entanglement can be achieved efficiently., Comment: Replaced with version similar to the published. Added journal reference.

Entanglement distillation aims at preparing highly entangled states out of a supply of weakly entangled pairs, using local devices and classical communication only. In this note we discuss the experimentally feasible schemes for optical continuous-variable entanglement distillation that have been presented in [D.E. Browne, J. Eisert, S. Scheel, and M.B. Plenio, Phys. Rev. A 67, 062320 (2003)] and [J. Eisert, D.E. Browne, S. Scheel, and M.B. Plenio, Annals of Physics (NY) 311, 431 (2004)]. We emphasize their versatility in particular with regards to the detection process and discuss the merits of the two proposed detection schemes, namely photo-detection and homodyne detection, in the light of experimental realizations of this idea becoming more and more feasible., Comment: 5 pages, 5 figures, contribution to conference proceedings

We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions. The approach is based on weighted graph states and superpositions thereof. These states allow for the efficient computation of all local observables (e.g. energy) and include states with diverging correlation length and unbounded multi-particle entanglement. As a demonstration we apply our approach to the Ising model on 1D, 2D and 3D square-lattices. We also present generalizations to higher spins and continuous-variable systems, which allows for the investigation of lattice field theories., Comment: 4 pages, 4 figures

We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respect to the rest of a general harmonic lattice system in the ground or a thermal state scale at most as the boundary area of the region. This area law is rigorously proven to hold true in non-critical harmonic lattice system of arbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape and states of nonzero temperature. For nearest-neighbor interactions - corresponding to the Klein-Gordon case - upper and lower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizing the findings of [Phys. Rev. Lett. 94, 060503 (2005)]. These higher dimensional analogues of the analysis of block entropies in the one-dimensional case show that under general conditions, one can expect an area law for the entanglement in non-critical harmonic many-body systems. The proofs make use of methods from entanglement theory, as well as of results on matrix functions of block banded matrices. Disordered systems are also considered. We moreover construct a class of examples for which the two-point correlation length diverges, yet still an area law can be proven to hold. We finally consider the scaling of classical correlations in a classical harmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment on a general relationship between criticality and area laws for the entropy of entanglement., Comment: 15 pages (RevTeX), 5 figures, minor changes

It is proven that the logarithmic negativity does not increase on average under positive partial transpose preserving (PPT) operation including subselection (a set of operations that incorporate local operations and classical communication (LOCC) as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is obtained from the negativity applying a concave function and is itself not a convex quantity this result is surprising as convexity is generally considered as describing the local physical process of losing information. The role of convexity and in particular its relation (or lack thereof) to physical processes is discussed in this context., Comment: Replaced with published version which includes one figure

We demonstrate that even under positive partial transpose preserving operations in an asymptotic setting GHZ and W states are not reversibly interconvertible. We investigate the structure of minimal reversible entanglement generating set (MREGS) for tri-partite states under positive partial transpose (ppt) preserving operations. We demonstrate that the set consisting of W and EPR states alone cannot be an MREGS. In this context we prove the surprising result that the relative entropy of entanglement can be strictly sub-additive for certain pure tri-partite states which is crucial to keep open the possibility that the set of GHZ-state and EPR states together constitute an MREGS under ppt-preserving operations., Comment: Minor changes in the presentation, proof of strict sub-additivity of relative entropy of entanglement for pure tri-partite states moved into main text, results unchanged

We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians corresponding to discrete versions of real free Klein-Gordon fields. For the ground state of the d-dimensional cubic harmonic lattice we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice analytically, without resorting to numerical means. We outline extensions of these results to longer ranged interactions, finite temperatures and for classical correlations in classical harmonic lattice systems. These findings further suggest that the tools of quantum information science may help in establishing results in quantum field theory that were previously less accessible., Comment: 4 pages, 1 figure. Replaced with published version

We study the entanglement dynamics of a system consisting of a large number of coupled harmonic oscillators in various configurations and for different types of nearest neighbour interactions. For a one-dimensional chain we provide compact analytical solutions and approximations to the dynamical evolution of the entanglement between spatially separated oscillators. Key properties such as the speed of entanglement propagation, the maximum amount of transferred entanglement and the efficiency for the entanglement transfer are computed. For harmonic oscillators coupled by springs, corresponding to a phonon model, we observe a non-monotonic transfer efficiency in the initially prepared amount of entanglement, i.e., an intermediate amount of initial entanglement is transferred with the highest efficiency. In contrast, within the framework of the rotating wave approximation (as appropriate e.g. in quantum optical settings) one finds a monotonic behaviour. We also study geometrical configurations that are analogous to quantum optical devices (such as beamsplitters and interferometers) and observe characteristic differences when initially thermal or squeezed states are entering these devices. We show that these devices may be switched on and off by changing the properties of an individual oscillator. They may therefore be used as building blocks of large fixed and pre-fabricated but programmable structures in which quantum information is manipulated through propagation. We discuss briefly possible experimental realisations of systems of interacting harmonic oscillators in which these effects may be confirmed experimentally., Comment: 20 pages, 17 figures, replaced with published version

We outline the basic questions that are being studied in the theory of entanglement. Following a brief review of some of the main achievements of entanglement theory for finite-dimensional quantum systems such as qubits, we will consider entanglement in infinite-dimensional systems. Asking for a theory of entanglement in such systems under experimentally feasible operations leads to the development of the theory of entanglement of Gaussian states. Results of this theory are presented and the tools that have been developed for it are applied to a number of problems., Comment: 14 pages, 4 figures. Invited contribution to the EQIS' 03 volume of the International Journal of Quantum Information. Technical problem with figures resolved

We study arrays of mechanical oscillators in the quantum domain and demonstrate how the motions of distant oscillators can be entangled without the need for control of individual oscillators and without a direct interaction between them. These oscillators are thought of as being members of an array of nano-electromechanical resonators with a voltage being applicable between neighboring resonators. Sudden non-adiabatic switching of the interaction results in a squeezing of the states of the mechanical oscillators, leading to an entanglement transport in chains of mechanical oscillators. We discuss spatial dimensions, Q-factors, temperatures and decoherence sources in some detail, and find a distinct robustness of the entanglement in the canonical coordinates in such a scheme. We also briefly discuss the challenging aspect of detection of the generated entanglement., Comment: 4 pages, 3 figures, new material added

We present an event-ready procedure that is capable of distilling Gaussian two-mode entangled states from a supply of weakly entangled states that have become mixed in a decoherence process. This procedure relies on passive optical elements and photon detectors distinguishing the presence and the absence of photons, but does not make use of photon counters. We identify fixed points of the iteration map, and discuss in detail its convergence properties. Necessary and sufficient criteria for the convergence to two-mode Gaussian states are presented. On the basis of various examples we discuss the performance of the procedure as far as the increase of the degree of entanglement and two-mode squeezing is concerned. Finally, we consider imperfect operations and outline the robustness of the scheme under non-unit detection efficiencies of the detectors. This analysis implies that the proposed protocol can be implemented with currently available technology and detector efficiencies., Comment: 14 pages, 12 figures

We present a protocol that allows the generation of a maximally entangled state between individual atoms held in spatially separate cavities. Assuming perfect detectors and neglecting spontaneous emission from the atoms, the resulting idealized scheme is deterministic. Under more realistic conditions, when the the atom-cavity interaction departs from the strong coupling regime, and considering imperfect detectors, we show that the scheme is robust against experimental inefficiencies and yields probabilistic entanglement of very high fidelity., Comment: 5 pages, 3 figures

We present a theory of entanglement transformations of Gaussian pure states with local Gaussian operations and classical communication. This is the experimentally accessible set of operations that can be realized with optical elements such as beam splitters, phase shifts and squeezers, together with homodyne measurements. We provide a simple necessary and sufficient condition for the possibility to transform a pure bipartite Gaussian state into another one. We contrast our criterion with what is possible if general local operations are available., Comment: 12 pages, 1 figure

We study the local implementation of POVMs when we require only the faithful reproduction of the statistics of the measurement outcomes for all initial states. We first demonstrate that any POVM with separable elements can be implemented by a separable super-operator, and develop techniques for calculating the extreme points of POVMs under a certain class of constraint that includes separability and PPT-ness. As examples we consider measurements that are invariant under various symmetry groups (Werner, Isotropic, Bell-diagonal, Local Orthogonal), and demonstrate that in these cases separability of the POVM elements is equivalent to implementability via LOCC. We also calculate the extrema of these classes of measurement under the groups that we consider, and give explicit LOCC protocols for attaining them. These protocols are hence optimal methods for locally discriminating between states of these symmetries. One of many interesting consequences is that the best way to locally discriminate Bell diagonal mixed states is to perform a 2-outcome POVM using local von Neumann projections. This is true regardless of the cost function, the number of states being discriminated, or the prior probabilities. Our results give the first cases of local mixed state discrimination that can be analysed quantitatively in full, and may have application to other problems such as demonstrations of non-locality, experimental entanglement witnesses, and perhaps even entanglement distillation., Comment: 15 pages, 1 figure. Title changed, and explanation improved in accordance with published verion (Phys. Rev. A.)

We investigate the properties of three entanglement measures that quantify the statistical distinguishability of a given state with the closest disentangled state that has the same reductions as the primary state. In particular, we concentrate on the relative entropy of entanglement with reversed entries. We show that this quantity is an entanglement monotone which is strongly additive, thereby demonstrating that monotonicity under local quantum operations and strong additivity are compatible in principle. In accordance with the presented statistical interpretation which is provided, this entanglement monotone, however, has the property that it diverges on pure states, with the consequence that it cannot distinguish the degree of entanglement of different pure states. We also prove that the relative entropy of entanglement with respect to the set of disentangled states that have identical reductions to the primary state is an entanglement monotone. We finally investigate the trace-norm measure and demonstrate that it is also a proper entanglement monotone., Comment: 7 pages, 1 figure

We introduce a protocol that maps finite-dimensional pure input states onto approximately Gaussian states in an iterative procedure. This protocol can be used to distill highly entangled bi-partite Gaussian states from a supply of weakly entangled pure Gaussian states. The entire procedure requires only the use of passive optical elements and photon detectors that solely distinguish between the presence and absence of photons., Comment: Some typographical errors corrected and discussions extended, title changed

Current methods of preparing maximally entangled states in the laboratory need an extremely accurate control of interaction times, requiring sophisticated experimental techniques. Here, we show that such precise control is not necessary when one utilizes short or weak interactions followed by measurements. We present a scheme for the probabilistic generation of Bell states in a pair of cavities, after each has interacted briefly with an atom. The advantage of this proposal, as compared to present schemes, is its low sensitivity to the exact values of experimental parameters such as atomic velocity and coupling strength, in fact, for a large range of parameters the fidelity of the Bell states generated remains close to unity., Comment: 8 pages, 6 figures, minor improvements to layout to improve readability, 1 reference added

We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT-operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this easily computable entanglement measure. As examples we discuss general Werner states and arbitrary bi-partite Gaussian states. Equipped with this result we then prove that for the anti-symmetric Werner state PPT-cost and PPT-entanglement of distillation coincide giving the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible., Comment: 4 pages, 1 figure

How mixed can one component of a bi-partite system be initially and still become entangled through interaction with a thermalized partner? We address this question here. In particular, we consider the question of how mixed a two-level system and a field mode may be such that free entanglement arises in the course of the time evolution according to a Jaynes-Cummings type interaction. We investigate the situation for which the two-level system is initially in mixed state taken from a one-parameter set, whereas the field has been prepared in an arbitrary thermal state. Depending on the particular choice for the initial state and the initial temperature of the quantised field mode, three cases can be distinguished: (i) free entanglement will be created immediately, (ii) free entanglement will be generated, but only at a later time different from zero, (iii) the partial transpose of the joint state remains positive at all times. It will be demonstrated that increasing the initial temperature of the field mode may cause the joint state to become distillable during the time evolution, in contrast to a non-distillable state at lower initial temperatures. We further assess the generated entanglement quantitatively, by evaluating the logarithmic negativity numerically, and by providing an analytical upper bound., Comment: 5 pages, 2 figures. Contribution to the proceedings of the 'International Conference on Quantum Information', Oviedo, July 13-18, 2002. Discusses sudden changes of entanglement properties in a dynamical quantum model

We investigate the entangling capability of passive optical elements, both qualitatively and quantitatively. We present a general necessary and sufficient condition for the possibility of creating distillable entanglement in an arbitrary multi-mode Gaussian state with the help of passive optical elements, thereby establishing a general connection between squeezing and the entanglement that is attainable by non-squeezing operations. Special attention is devoted to general two-mode Gaussian states, for which we provide the optimal entangling procedure, present an explicit formula for the attainable degree of entanglement measured in terms of the logarithmic negativity, and discuss several practically important special cases., Comment: 4 pages, 1 figure, replaced with published version

We study the entanglement properties of a closed chain of harmonic oscillators that are coupled via a translationally invariant Hamiltonian, where the coupling acts only on the position operators. We consider the ground state and thermal states of this system, which are Gaussian states. The entanglement properties of these states can be completely characterized analytically when one uses the logarithmic negativity as a measure of entanglement., Comment: 15 pages, 14 figures, minor typos corrected

We show that no distillation protocol for Gaussian quantum states exists that relies on (i) arbitrary local unitary operations that preserve the Gaussian character of the state and (ii) homodyne detection together with classical communication and postprocessing by means of local Gaussian unitary operations on two symmetric identically prepared copies. This is in contrast to the finite-dimensional case, where entanglement can be distilled in an iterative protocol using two copies at a time. The ramifications for the distribution of Gaussian states over large distances will be outlined. We also comment on the generality of the approach and sketch the most general form of a Gaussian local operation with classical communication in a bipartite setting., Comment: 4 pages, 1 colour figure. This paper is closely related to [J. Fiurasek, PRL 89, 137904 (2002), quant-ph/0204069] and to [G. Giedke and J.I. Cirac, PRA 66, 032316 (2002), quant-ph/0204085]. In particular, the latter paper arrives with independent methods at the general result that distillation is not possible for an arbitrary number of input copies and general Gaussian operations. The title of our contribution in its printed version has been changed by the editors of Physical Review Letters

We demonstrate that quantum information processing can be implemented with ions trapped in a far detuned optical cavity. For sufficiently large detuning the system becomes insensitive to cavity decay. Following recent experimental progress, this scheme can be implemented with currently available technology., Comment: 4 pages, 6 figures

We investigate the entanglement properties of the joint state of a distinguished quantum system and its environment in the quantum Brownian motion model. This model is a frequent starting point for investigations of environment-induced superselection. Using recent methods from quantum information theory, we show that there exists a large class of initial states for which no entanglement will be created at all times between the system of salient interest and the environment. If the distinguished system has been initially prepared in a pure Gaussian state, then entanglement is created immediately, regardless of the temperature of the environment and the non-vanishing coupling., Comment: 4 pages, 1 figure, revised version, new material added, title changed

An atom that couples to two distinct leaky optical cavities is driven by an external optical white noise field. We describe how entanglement between the light fields sustained by two optical cavities arises in such a situation. The entanglement is maximized for intermediate values of the cavity damping rates and the intensity of the white noise field, vanishing both for small and for large values of these parameters and thus exhibiting a stochastic-resonance-like behaviour. This example illustrates the possibility of generating entanglement by exclusively incoherent means and sheds new light on the constructive role noise may play in certain tasks of interest for quantum information processing., Comment: Second part of the paper rewritten, more discussion and some analytical results added