The entanglement of formation of bipartite quantum Gaussian state is investigated by means of local operation, which maps continuous variable state to a bipartite qubits system. A two parameters quantum Guassian state is introduced, the concurrence of its mapped qubits state is calculated to characterize the entanglement of the original state.
The entanglement of formation of bipartite quantum Gaussian state is investigated by means of local operation, which maps continuous variable state to a bipartite qubits system. A two parameters quantum Guassian state is introduced, the concurrence of its mapped qubits state is calculated to characterize the entanglement of the original state.
We present an approach to the analysis of entanglement in smooth bipartite continuous-variable states. One or both parties perform projective filterings via preliminary measurements to determine whether the system is located in some region of space; we study the entanglement remaining after filtering. For small regions, a two-mode system can be approximated by a pair of qubits and its entanglement fully characterized, even for mixed states. Our approach may be extended to any smooth bipartite pure state or two-mode mixed state, leading to natural definitions of concurrence and negativity densities. For Gaussian states both these quantities are constant throughout configuration space.
In this chapter we review the characterization of entanglement in Gaussian states of continuous variable systems. For two-mode Gaussian states, we discuss how their bipartite entanglement can be accurately quantified in terms of the global and local amounts of mixedness, and efficiently estimated by direct measurements of the associated purities. For multimode Gaussian states endowed with local symmetry with respect to a given bipartition, we show how the multimode block entanglement can be completely and reversibly localized onto a single pair of modes by local, unitary operations. We then analyze the distribution of entanglement among multiple parties in multimode Gaussian states. We introduce the continuous-variable tangle to quantify entanglement sharing in Gaussian states and we prove that it satisfies the Coffman-Kundu-Wootters monogamy inequality. Nevertheless, we show that pure, symmetric three-mode Gaussian states, at variance with their discrete-variable counterparts, allow a promiscuous sharing of quantum correlations, exhibiting both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. Finally, we investigate the connection between multipartite entanglement and the optimal fidelity in a continuous-variable quantum teleportation network. We show how the fidelity can be maximized in terms of the best preparation of the shared entangled resources and, viceversa, that this optimal fidelity provides a clearcut operational interpretation of several measures of bipartite and multipartite entanglement, including the entanglement of formation, the localizable entanglement, and the continuous-variable tangle.
We experimentally generate and tomographically characterize a mixed, genuinely non-Gaussian bipartite continuous-variable entangled state. By testing entanglement in 2$\ifmmode\times\else\texttimes\fi{}$2-dimensional two-qubit subspaces, entangled qubits are localized within the density matrix, which, first, proves the distillability of the state and, second, is useful to estimate the efficiency and test the applicability of distillation protocols. In our example, the entangled qubits are arranged in the density matrix in an asymmetric way, i.e., entanglement is found between diverse qubits composed of different photon number states, although the entangled state is symmetric under exchanging the modes.
Quantum entanglement is the key property that makes quantum information theory different from its classical counterpart and is also a valuable physical resource with massive potential for technological applications. However, our understanding of entanglement is still far from complete despite intense research activities. Like other physical resources, the first step towards exploiting them fully is to know how to quantify. There are many reasons to focus on the entanglement of continuous-variable states since the underlying degrees of freedom of physical systems carrying quantum information are frequently continuous, rather than discrete. Much of the effort has been concentrated on Gaussian states, because these are common as the ground or \nthermal states of optical modes. Within this framework, many interesting topics have been studied and some significant progress made. Nevertheless, non-Gaussian states are also extremely important; this is especially so in condensed-phase systems, where harmonic behaviour in any \ndegree of freedom is likely to be only an approximation. So far, there is little knowledge about the quantification of entanglement in non-Gaussian states. \nThis thesis aims to contribute to the active field of research in quantum entanglement by introducing a new approach to the analysis of entanglement, especially in continuous-variable states, and shows that it leads to the first systematic quantification of the (local) entanglement \nin arbitrary bipartite non-Gaussian states. By applying this local approach, many new insights can be gained. Notably, local entanglements of systems with smooth wavefunctions are fully characterised by the derived simple expressions, provided the wavefunction is known. The local (logarithmic) negativity of any two-mode mixed states can be directly computed from the closed-form formulae given. For multi-mode mixed states, this approach provides a scheme that \npermits much simpler numerical computation for quantifying entanglement than is generally possible from directly computing the full entanglement of the system.
Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.