We investigate entanglement between collective operators of two blocks of oscillators in an infinite linear harmonic chain. These operators are defined as averages over local operators (individual oscillators) in the blocks. On the one hand, this approach of "physical blocks" meets realistic experimental conditions, where measurement apparatuses do not interact with single oscillators but rather with a whole bunch of them, i.e., where in contrast to usually studied "mathematical blocks" not every possible measurement is allowed. On the other, this formalism naturally allows the generalization to blocks which may consist of several non-contiguous regions. We quantify entanglement between the collective operators by a measure based on the Peres-Horodecki criterion and show how it can be extracted and transferred to two qubits. Entanglement between two blocks is found even in the case where none of the oscillators from one block is entangled with an oscillator from the other, showing genuine bipartite entanglement between collective operators. Allowing the blocks to consist of a periodic sequence of subblocks, we verify that entanglement scales at most with the total boundary region. We also apply the approach of collective operators to scalar quantum field theory.
We study robustness of genuine multipartite entanglement for a system of three qubits under collective dephasing. Using a computable entanglement monotone for multipartite systems, we found that almost every state is quite robust under this type of decoherence. We analyze random states and weighted graph states at infinity and found all of them to be genuinely entangled.
We show how shared entanglement, together with classical communication and local quantum operations, can be used to perform an arbitrary collective quantum operation upon N spatially-separated qubits. A simple teleportation-based protocol for achieving this, which requires 2(N-1) ebits of shared, bipartite entanglement and 4(N-1) classical bits, is proposed. In terms of the total required entanglement, this protocol is shown to be optimal for even N in both the asymptotic limit and for `one-shot' applications.
We investigate the second-order non-linear interaction as a means to generate entanglement between fields of differing wavelengths. And show that perfect entanglement can, in principle, be produced between the fundamental and second harmonic fields in these processes. Neither pure second harmonic generation, nor parametric oscillation optimally produce entanglement, such optimal entanglement is rather produced by an intermediate process. An experimental demonstration of these predictions should be imminently feasible.
We present a method of generating collective multi-qubit entanglement via global addressing of an ion chain following the guidelines of the Tavis-Cummings model, where several qubits are coupled to a collective motional mode. We show that a wide family of Dicke states and irradiant states can be generated by single global laser pulses, unitarily or helped with suitable postselection techniques.
In this paper, we investigate how to reduce the number of measurement configurations needed for sufficiently precise entanglement quantification. Instead of analytical formulae, we employ artificial neural networks to predict the amount of entanglement in a quantum state based on results of collective measurements (simultaneous measurements on multiple instances of the investigated state). This approach allows us to explore the precision of entanglement quantification as a function of measurement configurations. For the purpose of our research, we consider general two-qubit states and their ne…
In this paper, an optimal scheme of four-level quantum teleportation and swapping of quantum entanglement is given. We construct a complete orthogonal basis of the bipartite ququadrit systems. Using this basis, the four-level quantum teleportation and swapping can be achieved according to the standard steps. In addition, associate the above bases with the unextendible product bases and the exact entanglement bases, we prove that in the $2\times 2\times 2$ systems or $3\times 3$ systems the collective translations of multipartite quantum entanglement can be realized. PACC numbers: 03.67.Mn, 03.…
We study the boson-parafermion entanglement of the parasupersymmetric coherent states of the harmonic oscillator and derive the degree of entanglement in terms of the concurrence. The conditions for obtaining the maximal entanglement is also examined, and it is shown that in the usual supersymmetry situation we can obtain maximally entangled Bell states.
Known entanglement purification protocols for mixed states use collective measurements on several copies of the state in order to increase the entanglement of some of them. We address the question of whether it is possible to purify the entanglement of a state by processing each copy separately. While this is possible for pure states, we show that this is impossible, in general, for mixed states. The importance of this result both conceptually and for experimental realization of purification is discussed. We also give explicit invariants of an entangled state of two qubits under local actions …
We present an efficient quantum entanglement distribution over an arbitrary collective-noise channel. The basic idea in the present scheme is that two parties in quantum communication first transmit the entangled states in the frequency degree of freedom which suffers little from the noise in an optical fiber. After the two parties share the photon pairs, they add some operations and equipments to transfer the frequency entanglement of pairs into the polarization entanglement with the success probability of 100\%. Finally, they can get maximally entangled polarization states with polarization …
We study the dynamics of genuine multipartite entanglement for quantum systems upto four qubits interacting with general collective dephasing process. Using a computable entanglement monotone for multipartite systems, we observe the feature of freezing dynamics of genuine entanglement for three and four qubits entangled states. We compare the dynamics with that of random states and find that most states exibit this feature. We then study the effects of collective dephasing on genuine nonlocality and find out that although quantum states remain genuinely entangled yet their genuine nonlocality …
We study the harmonic entanglement and squeezing in a two-mode radiation produced in a degenerate parametric down conversion process coupled to a two-mode vacuum reservoir employing the linearization procedure. It is found that there is a quadrature entanglement between the harmonically related fundamental and second-harmonic modes and the superimposed radiation exhibits a significant two-mode squeezing. The entanglement exits even when there is no two-mode squeezing, since the correlation leading to these phenomena are essentially different. In addition, the more the external coherent light i…
We investigate the entanglement properties of thermal states of the harmonic lattice in one, two and three dimensions. We establish the value of the critical temperature for entanglement between neighbouring sites and give physical reasons. Further sites are shown to be entangled only due to boundary effects. Other forms of entanglement are addressed in the second part of the paper by using the energy as witness of entanglement. We close with a comprehensive diagram showing the different phases of entanglement versus complete separability and propose techniques to swap and tune entanglement ex…
We investigate the dynamics of entanglement and nonlocality for multipartite quantum systems under collective dephasing. Using an exact and computable measure for genuine entanglement, we demonstrate the possibility of a non trivial phenomenon of time-invariant entanglement for multipartite quantum systems. We find that for four qubits, there exist quantum states, which are changing continously nevertheless their genuine entanglement remains constant. Based on our numerical results, we conjecture that there is no evidence of time-invariant entanglement for quantum states of three qubits. We po…
We study the ground-state entanglement of one-dimensional harmonic chains that are coupled to each other by a collective interaction as realized e.g. in an anisotropic ion crystal. Due to the collective type of coupling, where each chain interacts with every other one in the same way,the total system shows critical behavior in the direction orthogonal to the chains while the isolated harmonic chains can be gapped and non-critical. We derive lower and most importantly upper bounds for the entanglement,quantified by the von Neumann entropy, between a compact block of oscillators and its environm…
We analyze the quantum entanglement between two interacting atoms trapped in a spherical harmonic potential. At ultra-cold temperature, ground state entanglement is generated by the dominated s-wave interaction. Based on a regularized pseudo-potential Hamiltonian, we examine the quantum entanglement by performing the Schmidt decomposition of low-energy eigenfunctions. We indicate how the atoms are paired and quantify the entanglement as a function of a modified s-wave scattering length inside the trap.
We describe an experimentally straightforward method for preparing an entangled W state of up to 100 qubits. Our repeat-until-success protocol relies on detection of single photons from collective spontaneous emission in free space. Our method allows entanglement preparation in a wide range of qubit implementations that lack entangling qubit-qubit interactions. We give detailed numerical examples for entanglement of neutral atoms in optical lattices and of nitrogen-vacancy centres in diamond. The simplicity of our method should enable preparation of mesoscopic entangled states in a number of p…
A cogent theory of collective multipole-like quantum correlations in symmetric multiqubit states is presented by employing SO(3) irreducible spherical tensor representation. An arbitrary bipartite division of this system leads to a family of inequalities to detect entanglement involving averages of these tensors expressed in terms of the total system angular momentum operator. Implications of this theory to the quantum nature of multipole-like correlations of all orders in the Dicke states are deduced. A selected set of examples illustrate these collective tests. Such tests detect entanglement…
The limitations on the production and profitability of entanglement in a harmonic chain under strong driving are considered. We report on the limits of attainable entanglement for a given set of squeezings of the eigenmodes, showing that the higher the entanglement the more oscillatory and thus less easy to profit from. We also comment on propagation properties of entanglement, discussing the role of fast rotating terms and illustrating several issues with the example of a sudden switch of the coupling.
We investigate quantum correlations of the $N$-qubit states via a collective pseudo-spin interaction ($\propto J_y^2$) on arbitrary pure separable states for a given interval of time. Based on this dynamical generation of the $N$-qubit maximal entangled states, a quantum secret sharing protocol with $N$ continuous classical secrets is developed.
We study multipartite entanglement in non-equilibrium quantum phase transition (NEQPT) attainable in a coherently driven atomic ensemble undergoing collective decay. The NEQPT arises in the steady state of the system as the drive field strength is varied in comparison with the decay parameter. A close connection is observed between the critical behavior and the multi-partite entanglement in the system determined via the von Neumann entropy. The derivative of the von Neumann entropy shows Lambda-type behavior typical of the second order phase transitions. We further show that the calculated bi-…
The entanglement between a Pauli-like two-level system and a quantum harmonic oscillator enhanced by an interaction between them and a $δ$-pulse sequence is studied, with the decoherence due to their coupling with a Markovian bath. Without the Markovian bath, the entanglement is enhanced to maximum possible values. With the Markovian bath, the entanglement is enhanced up to some time and then dissipated, with the system in thermal equilibrium with the Markovian bath after a very long time. The time for achieving the maximum entanglement shows discontinuous jumps over the parameters of decohere…
We present an economical setup for faithful entanglement sharing against collective noise. It is composed of polarizing beam splitters, half wave plates, polarization independent wavelength division multiplexers, and frequency shifters. An arbitrary qubit error on the polarization state of each photon in a multi-photon system caused by the noisy channel can be rejected, without resorting to additional qubits, fast polarization modulators, and nondestructive quantum nondemolition detectors. Its success probability is in principle 100%, which is independent of the noise parameters, and it can be…
For a given pure state of a composite quantum system we analyze the product of its projections onto a set of locally orthogonal separable pure states. We derive a bound for this product analogous to the entropic uncertainty relations. For bipartite systems the bound is saturated for maximally entangled states and it allows us to construct a family of entanglement measures, we shall call collectibility. As these quantities are experimentally accessible, the approach advocated contributes to the task of experimental quantification of quantum entanglement, while for a three-qubit system it is cap…
We investigate quantum entanglement between two (spin-1/2) fermions inside a cylindrical harmonic trap, making use of the von Neumann entropy for the reduced single particle density matrix as the pure state entanglement measure. We explore the dependence of pair entanglement on the geometry and strength of the trap and on the strength of the pairing interaction over the complete range of the effective BCS to BEC crossover. Our result elucidates an interesting connection between our model system of two fermions and that of two interacting bosons.
We analyse a nonlinear optical system which uses cascaded nonlinearities to produce both second and fourth harmonic outputs from an input field at the fundamental frequency. Using fully quantum equations of motion, we show that the system produces quadrature squeezed outputs which exhibit bipartite entanglement, EPR-steering, and asymmetric steering across a two octave frequency range.
We investigate the stationary entanglement and stationary nonlocality of two qubits collectively interacting with a common thermal environment. We assume two qubits are initially in Werner state or Werner-like state, and find that thermal environment can make two qubits become stationary nonlocality. The analytical relations among average thermal photon number of the environment, entanglement and nonlocality of two qubits are given in details. It is shown that the fraction of Bell singlet state plays a key role in the phenomenon that the common thermal reservoir can enhance the entanglement of…
We introduce a general scheme to detect various multiparticle entanglement structures from global non-permutationally invariant observables. In particular, we derive bounds on the variance of non-permutationally invariant and collective operators for the verification of $k$-party entanglement. For a family of observables related to the spin structure factor, we give quantitative bounds on entanglement that are independent of the total number of particles. We introduce highly non-symmetric states with genuine multipartite entanglement that is verifiable with the presented technique and discuss …
We compare the bipartite entanglement and EPR-steering properties of the two different schemes which produce third harmonic optical fields from an input field at the fundamental frequency. The first scheme uses second harmonic cascaded with sum-frequency generation while the second uses triply degenerate four-wave mixing, also known as direct third harmonic generation. We examine and compare the two schemes in both the travelling wave and intra-cavity configurations. We find that both schemes produce continuous-variable bipartite entanglement and EPR-steering. The direct scheme produces a grea…
We analyze rigorously the dynamics of the entanglement between two qubits which interact only through collective and local environments. Our approach is based on the resonance perturbation theory which assumes a small interaction between the qubits and the environments. The main advantage of our approach is that the expressions for (i) characteristic time-scales, such as decoherence, disentanglement, and relaxation, and (ii) observables are not limited by finite times. We introduce a new classification of decoherence times based on clustering of the reduced density matrix elements. The charact…
We apply the Walborn [Walborn S P, Taketani B G, Salles A, Toscano F and de Matos Filho R L 2009 Entropic Entanglement Criteria for Continuous Variables Phys. Rev. Lett. 103 160505] Shannon Entropic Entanglement Criterion (SEEC), to simple harmonic oscillators, a system typically discussed in undergraduate quantum mechanics courses. In particular, we investigate the entanglement of a system of coupled harmonic oscillators. A simple form of entanglement criterion in terms of their interaction is found. It is shown that a pair of interacting ground state coupled oscillators are more likely to be…
Self-organization effects related to light amplification in the collective atomic recoil laser system with the driven atoms confined in a harmonic trap are investigated further. In the dispersive parametric region, our study reveals that the spontaneously formed structures in the phase space contributes an important role to the light amplification of the probe field under the atomic motion being modified by the trap.
We address the presence of non-distillable (bound) entanglement in natural many-body systems. In particular, we consider standard harmonic and spin-1/2 chains, at thermal equilibrium and characterized by few interaction parameters. The existence of bound entanglement is addressed by calculating explicitly the negativity of entanglement for different partitions. This allows to individuate a range of temperatures for which no entanglement can be distilled by means of local operations, despite the system being globally entangled. We discuss how the appearance of bound entanglement can be linked t…
The entanglement properties of the phase transition in a two dimensional harmonic lattice, similar to the one observed in recent ion trap experiments, are discussed both, for finite number of particles and thermodynamical limit. We show that for the ground state at the critical value of the trapping potential two entanglement measures, the negativity between two neighbouring sites and the block entropy for blocks of size 1, 2 and 3, change abruptly. Entanglement thus indicates quantum phase transitions in general; not only in the finite dimensional case considered in [Phys. Rev. Lett. {\bf 93}…
We study the entanglement evolution between two harmonic oscillators having different free frequencies each leaking into an independent bath. We use an exact solution valid in the weak coupling limit and in the short time non-Markovian regime. The reservoirs are identical and characterized by an Ohmic spectral distribution with Lorents-Drude cut-off. This work is an extension of the case reported in [Phys. Rev. A 80, 062324 (2009)] where the oscillators have the same free frequency.
We introduce a special class of truncated Weyl-Heisenberg algebra and discuss the corresponding Hilbertian and analytical representations. Subsequently, we study the effect of a quantum network of beam splitting on coherent states of this nonlinear class of harmonic oscillators. We particularly focus on quantum networks involving one and two beam splitters and examine the degree of bipartite as well as tripartite entanglement using the linear entropy.
The exact dynamics of the entanglement between two harmonic modes generated by an angular momentum coupling is examined. Such system arises when considering a particle in a rotating anisotropic harmonic trap or a charged particle in a fixed harmonic potential in a magnetic field, and exhibits a rich dynamical structure, with stable, unstable and critical regimes according to the values of the rotational frequency or field and trap parameters. Consequently, it is shown that the entanglement generated from an initially separable gaussian state can exhibit quite distinct evolutions, ranging from …
Entanglement between light and matter combines the advantage of long distance transmission of photonic qubits with the storage and processing capabilities of atomic qubits. To distribute photonic states efficiently over long distances several schemes to encode qubits have been investigated -- time-bin encoding being particularly promising due to its robustness against decoherence in optical fibers. Here, we demonstrate the generation of entanglement between a photonic time-bin qubit and a single collective atomic spin excitation (spin-wave) in a cold atomic ensemble, followed by the mapping of…
We examine the entanglement induced by an angular momentum coupling between two harmonic systems. The Hamiltonian corresponds to that of a charged particle in a uniform magnetic field in an anisotropic quadratic potential, or equivalently, to that of a particle in a rotating quadratic potential. We analyze both the vacuum and thermal entanglement, obtaining analytic expressions for the entanglement entropy and negativity through the gaussian state formalism. It is shown that vacuum entanglement diverges at the edges of the dynamically stable sectors, increasing with the angular momentum and sa…
A general and in principle exact approach for the continuous variable entanglement in a system of coupled harmonic oscillators in contact with a thermal bath is formulated. This allows a generalization to describe entanglement's existence between two sites in any system of this kind. It is employed a method of simultaneous diagonalization of two quadratic forms to obtain the uncoupled quantum Hamiltonian. Making use of the transformations that uncouple the system, the covariance matrix for two positions is constructed, and through the positive partial transpose criterion (PPT-criterion), the c…
A common environment acting on a pair of qubits gives rise to a plethora of different phenomena, such as the generation of qubit-qubit entanglement, quantum synchronization and subradiance. Here we define time-independent figures of merit for entanglement generation, quantum synchronization and subradiance, and perform an extensive analytical and numerical study of their dependence on model parameters. We also address a recently proposed measure of the collectiveness of the dynamics driven by the bath, and find that it almost perfectly witnesses the behavior of entanglement generation. Our res…
For both cases with and without interactions, bipartite entanglement of two-fermions from a Fermi gas in a trap is investigated. We show how the entanglement depends on the locations of the two fermions and the total number of particles. Fermions at the verge of trap have longer entanglement distance (beyond it, the entanglement disappears) than those in the center. We derive a lower limitation to the average overlapping for two entangled fermions in the BCS ground state, it is shown to be $\sqrt{Q/2M}$, a function of Cooper pair number $Q$ and total number of occupied energy levels $M$.
We describe a new experimental approach to probabilistic atom-photon (signal) entanglement. Two qubit states are encoded as orthogonal collective spin excitations of an unpolarized atomic ensemble. After a programmable delay, the atomic excitation is converted into a photon (idler). Polarization states of both the signal and the idler are recorded and are found to be in violation of the Bell inequality. Atomic coherence times exceeding several microseconds are achieved by switching off all the trapping fields - including the quadrupole magnetic field of the magneto-optical trap - and zeroing o…
We address the dynamics induced by collective atomic recoil in a Bose-Einstein condensate in presence of radiation losses and atomic decoherence. In particular, we focus on the linear regime of the lasing mechanism, and analyze the effects of losses and decoherence on the generation of entanglement. The dynamics is that of three bosons, two atomic modes interacting with a single-mode radiation field, coupled with a bath of oscillators. The resulting three-mode dissipative Master equation is solved analytically in terms of the Wigner function. We examine in details the two complementary limits …
We propose a scheme to realize entanglement swapping via superradiance, entangling two distant cavities without a direct interaction. The successful Bell-state-measurement outcomes are performed naturally by the electromagnetic reservoir, and we show how, using a quantum trajectory method, the non-local properties of the state obtained after the swapping procedure can be verified by the steering inequality. Furthermore, we discuss how the unsuccessful measurement outcomes can be used in an experiment of delayed-choice entanglement swapping. An extension of testing the quantum steering inequali…
We study the structure of vacuum entanglement for two complimentary segments of a linear harmonic chain, applying the modewise decomposition of entangled gaussian states discussed in \cite {modewise}. We find that the resulting entangled mode shape hierarchy shows a distinctive layered structure with well defined relations between the depth of the modes, their characteristic wavelength, and their entanglement contribution. We re-derive in the strong coupling (diverging correlation length) regime, the logarithmic dependence of entanglement on the segment size predicted by conformal field theory…
We investigate a simple arrangement of coupled harmonic oscillators which brings out some interesting effects concerning creation of entanglement. It is well known that if each member in a linear chain of coupled harmonic oscillators is prepared in a ``classical state'', such as a pure coherent state or a mixed thermal state, no entanglement is created in the rotating wave approximation. On the other hand, if one of the oscillators is prepared in a nonclassical state (pure squeezed state, for instance), entanglement may be created between members of the chain. In the setup considered here, we …
A simple scheme is presented for achieving effectively maximal pure-state entanglement between non-interacting atoms through purely collective decay and controlled symmetry breaking. The scheme requires no measurements or feedback or even knowledge of the initial states of the atoms. It relies on breaking the symmetry of the system Hamiltonian to ensure the existence of a unique attractive steady state and minimal control to achieve almost perfect overlap of this steady state with the maximally entangled singlet state. We demonstrate how our scheme can be implemented for two qubits encoded in …
Maximal entangled states (MES) provide a basis to two d-dimensional particles Hilbert space, d=prime $\ne 2$. The MES forming this basis are product states in the collective, center of mass and relative, coordinates. These states are associated (underpinned) with lines of finite geometry whose constituent points are associated with product states carrying Mutual Unbiased Bases (MUB) labels. This representation is shown to be convenient for the study of the Mean King Problem and a variant thereof, termed Tracking the King which proves to be a novel quantum communication channel. The main topics…
The paper reports on experimental diagnostics of entanglement swapping protocol by means of collective entanglement witness. Our approach is suitable to detect disturbances occurring in the preparation of quantum states, quantum communication channel and imperfect Bell-state projection. More specifically we demonstrate that our method can distinguish disturbances such as depolarization, phase-damping, amplitude-damping and imperfect Bell-state measurement by observing four probabilities and estimating collective entanglement witness. Since entanglement swapping is a key procedure for quantum r…
The state of a quantum system, consisting of two distinct subsystems, is called separable if it can be prepared by two distant experimenters who receive instructions from a common source, via classical communication channels. A necessary condition is derived and is shown to be more sensitive than Bell's inequality for detecting quantum inseparability. Moreover, collective tests of Bell's inequality (namely, tests that involve several composite systems simultaneously) may sometimes lead to a violation of Bell's inequality, even if the latter is satisfied when each composite system is tested sep…
Bipartite and global entanglement are analyzed for the ground state of a system of $N$ spin 1/2 particles interacting via a collective spin-spin coupling described by the Lipkin-Meshkov-Glick (LMG) Hamiltonian. Under certain conditions which includes the special case of a super-symmetry, the ground state can be constructed analytically. In the case of an anti-ferromagnetic coupling and for an even number of particles this state undergoes a smooth crossover as a function of the continuous anisotropy parameter $γ$ from a separable ($γ=\infty $) to a maximally entangled many-particle state ($γ=0$…
We discuss the relation between entanglement and criticality in translationally invariant harmonic lattice systems with non-randon, finite-range interactions. We show that the criticality of the system as well as validity or break-down of the entanglement area law are solely determined by the analytic properties of the spectral function of the oscillator system, which can easily be computed. In particular for finite-range couplings we find a one-to-one correspondence between an area-law scaling of the bi-partite entanglement and a finite correlation length. This relation is strict in the one-d…
In the present paper we study the entanglement properties of thermal (a.k.a. Gibbs) states of quantum harmonic oscillator systems as functions of the Hamiltonian and the temperature. We prove the physical intuition that at sufficiently high temperatures the thermal state becomes fully separable and we deduce bounds on the critical temperature at which this happens. We show that the bound becomes tight for a wide class of Hamiltonians with sufficient translation symmetry. We find, that at the crossover the thermal energy is of the order of the energy of the strongest normal mode of the system a…
We study the entanglement dynamics of the two two-level atoms coupling with a single-mode polarized cavity field after incorporating the atomic centers of mass classical harmonic vibrations with micro amplitudes and low frequencies. We propose a quantitative vibrant factor to modify the concurrence of the two atoms states. When the vibrant frequencies are very low, we obtain that: (i) the factor depends on the relative vibrant displacements and the initial phases rather than the absolute amplitudes, and reduces the concurrence to three orders of magnitude; (ii) the concurrence increases with t…
Maximal entangled states (MES) provide a basis to 2d-dimensional particles Hilbert space, d=prime $\ne2$. These states allow generalization of the Mean King Problem. The states may be viewed as build of points each underpins a product state carrying a mutual unbiased bases (MUB) label or, alternatively, as product states labeled with center of mass and relative coordinates. The coordinate-like label of the center of mass and the momentum-like of the relative coordinates provides a MES account of the Hilbert space in close analogy with the single particle phase space coordinates.
