Methods for Derivation of Density Matrix of Arbitrary Multi-Mode Gaussian States from Its Phase Space Representation

We present a method for derivation of the density matrix of an arbitrary multi-mode continuous variable Gaussian entangled state from its phase space representation.An explicit computer algorithm is given to reconstruct the density matrix from Gaussian covariance matrix and quadrature average values.As an example,we apply our method to the derivation of three-mode symmetric continuous variable entangled state.Our method can be used to analyze the entanglement and correlation in continuous variable quantum network with multi-mode quantum entanglement states.     

Extremality of Gaussian quantum states

We investigate Gaussian quantum states in view of their exceptional role within the space of all continuous variables states. A general method for deriving extremality results is provided and applied to entanglement measures, secret key distillation and the classical capacity of bosonic quantum channels. We prove that for every given covariance matrix the distillable secret key rate and the entanglement, if measured appropriately, are minimized by Gaussian states. This result leads to a clearer picture of the validity of frequently made Gaussian approximations. Moreover, it implies that Gaussian encodings are optimal for the transmission of classical information through bosonic channels, if the capacity is additive.     

Bound entangled Gaussian states

We discuss the entanglement properties of bipartite states with Gaussian Wigner functions. For the separability, and the positivity of the partial transpose, we establish explicit necessary and sufficient criteria in terms of the covariance matrix of the state. It is shown that, for systems composed of a single oscillator for Alice and an arbitrary number for Bob, positivity of the partial transpose implies separability. However, this implication fails with two oscillators on each side, as we show by constructing a five parameter family of bound entangled Gaussian states.     

Gaussian entanglement of symmetric two-mode Gaussian states

A Gaussian degree of entanglement for a symmetric two-mode Gaussian state can be defined as its distance to the set of all separable two-mode Gaussian states. The principal property that enables us to evaluate both Bures distance and relative entropy between symmetric two-mode Gaussian states is the diagonalization of their covariance matrices under the same beam-splitter transformation. The multiplicativity property of the Uhlmann fidelity and the additivity of the relative entropy allow one to finally deal with a single-mode optimization problem in both cases. We find that only the Bures-distance Gaussian entanglement is consistent with the exact entanglement of formation.     

Steering Witness and Steering Criterion of Gaussian States

Quantum steering is an important quantum resource, which is intermediate between entanglement and Bell nonlocality. In this paper, we study steering witnesses for Gaussian states in continuous-variable systems. We give a definition of steering witnesses by covariance matrices of Gaussian states, and then obtain a steering criterion by steering witnesses to detect steerability of any (m+n)-mode Gaussian states. In addition, the conditions for two steering witnesses to be comparable and the optimality of steering witnesses are also discussed.     

Evaluation of entanglement of CV-pair

Practical realizations of qubits and systems of qubits are implement by particles with states from infinite-dimensional space, such as space of wave functions for coordinate representation of quantum mechanics. Properties of the entangled states of continuous variables differ from such for discrete variables. That occurs due to existence of unlimited operators of observables and their uncertainties. In this work we used and extended idea of entanglement estimation for the case of continuous variables with the help of covariance matrix norm. Well-known concurrence was calculated with the help of covariance of observables. The goal of this paper is to check if it is possible to apply this idea to continuous variables. We consider coherent electron pair as the object of our investigation. We also consider coordinate and momentum of coherent electron pair as basis of observables. In this work, a new characteristic of entanglement is proposed. We evaluated that using correlation function instead of covariance. For the small distances between particles, it coincides with covariance matrix norm but for the big distances, it remains restricted.     

Minimal set of local measurements and classical communication for two-mode Gaussian state entanglement quantification

We develop the minimal requirements for the complete entanglement quantification of an arbitrary two-mode bipartite Gaussian state via local measurements and a classical communication channel. The minimal set of measurements is presented as a reconstruction protocol of local covariance matrices and no previous knowledge of the state is required but its Gaussian character. The protocol becomes very simple mostly when dealing with Gaussian states transformed to its standard form, since photocounting or intensity measurements define the whole set of entangled states. In addition, conditional on some prior information, the protocol is also useful for a complete global state reconstruction.     

Equivalence between entanglement and the optimal fidelity of continuous variable teleportation

We devise the optimal form of Gaussian resource states enabling continuous-variable teleportation with maximal fidelity. We show that a nonclassical optimal fidelity of N-user teleportation networks is necessary and sufficient for N-party entangled Gaussian resources, yielding an estimator of multipartite entanglement. The entanglement of teleportation is equivalent to the entanglement of formation in a two-user protocol, and to the localizable entanglement in a multiuser one. Finally, we show that the continuous-variable tangle, quantifying entanglement sharing in three-mode Gaussian states, is defined operationally in terms of the optimal fidelity of a tripartite teleportation network.     

Relative entropy as a measure of entanglement for Gaussian states

In this paper, we derive an explicit analytic expression of the relative entropy between two general Gaussian states. In the restriction of the set for Gaussian states and with the help of relative entropy formula and Peres--Simon separability criterion, one can conveniently obtain the relative entropy entanglement for Gaussian states. As an example, the relative entanglement for a two-mode squeezed thermal state has been obtained.     

Control of purity and entanglement for two spatially two-separated qubits via phase damping

Control of purity and entanglement of two two-qubits dispersively coupled to a field with a reservoir are investigated.Initially the qubits are entangled,while the field is either in a coherent state or a statistical mixture of two coherent states.For an alternative entanglement measure we calculate the negativity of the eigenvalues of the partially transposed density matrix.A measure related to the mutual entropy,namely the index of entropy,is employed to measure the entanglement.Its results agree well with the negativity.It is found that the entanglement and purity have strong sensitivity to phase damping.The asymptotic behaviour of the states of the field,the two two-qubits,and the total system fall into mixed states.     

Generic entanglement and standard form for N-mode pure Gaussian states

We investigate the correlation structure of pure N-mode Gaussian resources which can be experimentally generated by means of squeezers and beam splitters, whose entanglement properties are generic. We show that those states are specified (up to local unitaries) by N(N-1)/2 parameters, corresponding to the two-point correlations between any pair of modes. Our construction yields a practical scheme to engineer such generic-entangled N-mode pure Gaussian states by linear optics. We discuss our findings in the framework of Gaussian matrix product states of harmonic lattices, raising connections with entanglement frustration and the entropic area law.     

Broadening of the lowest Landau level by a Gaussian random potential with an arbitrary correlation length: an efficient continued-fraction approach

For an electron in the Euclidean plane subjected to a perpendicular constant magnetic field and a homogeneous Gaussian random potential with a Gaussian covariance function we approximate the averaged density of states restricted to the lowest Landau level. To this end, we extrapolate the first nine coefficients of the underlying continued fraction consistently with the coefficients’ high-order asymptotics. The latter derives from the known asymptotic decay of the density of states in the tails. We thus achieve on the one hand a reliable extension of Wegner’s exact result [Z. Phys. B 51, 279 (1983)] for the delta-correlated case to the physically more relevant case of a non-zero correlation length. On the other hand, we have thereby found a paragon for the power of continued-fraction expansions for designing approximations to spectral densities.     

Spin Fidelity for Three-Qubit Greenberger-Horne-Zeilinger and W States Under Lorentz Transformations

Constructing the reduced density matrix for a system of three massive spin- \frac12\frac{1}{2} particles described by a wave packet with Gaussian momentum distribution and a spin part in the form of GHZ or W state, the fidelity for the spin part of the system is investigated from the viewpoint of moving observers in the jargon of special relativity. Using a numerical approach, it turns out that by increasing the boost speed, the spin fidelity decreases and reaches to a non-zero asymptotic value that depends on the momentum distribution and the amount of momentum entanglement.     

Quantum and classical correlations in quantum Brownian motion

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 nonvanishing coupling.     

Death of entanglement and purity in a two qubits–field system induced by phase damping

We investigate the death of entanglement and the purity loss of a two qubits–field system in the dispersive regime with a reservoir. For an alternative entanglement measure, we calculate the negativity of the eigenvalues of a partially transposed density matrix and compare it with the mutual entropy. A new measure related to the mutual entropy, namely, the index of entropy, is proposed to measure the degree of entanglement, and this agrees well with the negativity. We found that the entanglement has a strong sensitivity to the phase damping. The asymptotic behavior of the field states, the two qubits, and the total system fall into a mixed state. We treat the phenomena of death of entanglement and purity as they arise from the effect of phase damping.     

Entangling two single-mode Gaussian states by use of a beam splitter

Considering two light beams which are in general single-mode Gaussian states and incident on input ports of an ideal beam splitter, respectively, this paper investigates how separability and entanglement of the output lights depend on degrees of nonclassicality and purities of the input states. The minimum and maximum amounts of attainable entanglement in the output state are found.