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arXiv:0711.3735 · Physical Review A

Local entanglement of multidimensional continuous-variable systems

Ho-Chih Lin, Andrew J. Fisher

Published Jul 25, 2008• 1 citation
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Abstract

We study the local entanglement remaining after filtering operations corresponding to imperfect measurements performed by one or both parties, such that the parties can only determine whether or not the system is located in some region of space. The local entanglement in pure states of general bipartite multidimensional continuous-variable systems can be completely determined through simple expressions. We apply our approach to semiclassical Wentzel-Kramers-Brillouin-Jeffreys (WKB) systems, multidimensional harmonic oscillators, and a hydrogen atom as three examples.

DOI
10.1103/physreva.78.012349

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We study the local entanglement remaining after filtering operations corresponding to imperfect measurements performed by one or both parties, such that the parties can only determine whether or not the system is located in some region of space. The local entanglement in pure states of general bipartite multidimensional continuous-variable systems can be completely determined through simple expressions. We apply our approach to semiclassical Wentzel-Kramers-Brillouin-Jeffreys (WKB) systems, multidimensional harmonic oscillators, and a hydrogen atom as three examples.
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.
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Quantum correlation provides a promising measure beyond entanglement. Here, we propose a necessary and sufficient condition for nonzero quantum correlation in continuous variable systems, which is simple and easy to perform in terms of a marker Q r . In order to get this condition, we introduce continuous-variable local orthogonal bases of the operator space, which are generalized from the orthogonal basis sets in local operator space for discrete variables. Based on this, we obtain the marker Q r for all bipartite continuous variable states, and provide several examples including two-mode Gaussian and non-Gaussian states. Our result may provide a candidate for quantum correlation measures, and can be measured by designed quantum circuits.
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In the quest to completely describe entanglement in the general case of a finite number of parties sharing a physical system of finite-dimensional Hilbert space an entanglement magnitude is introduced for its pure and mixed states: robustness. It corresponds to the minimal amount of mixing with locally prepared states which washes out all entanglement. It quantifies in a sense the endurance of entanglement against noise and jamming. Its properties are studied comprehensively. Analytical expressions for the robustness are given for pure states of two-party systems, and analytical bounds for mixed states of two-party systems. Specific results are obtained mainly for the qubit-qubit system (qubit denotes quantum bit). As by-products local pseudomixtures are generalized, a lower bound for the relative volume of separable states is deduced, and arguments for considering convexity a necessary condition of any entanglement measure are put forward.