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arXiv:quant-ph/0605154 · Physical Review A

Conditions for multipartite continuous-variable entanglement

Evgeny V Shchukin, W. Vogel

Published Sep 7, 2006• 40 citations
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Abstract

Necessary and sufficient observable conditions for the non-negativity of all partial transpositions of multimode quantum states are derived. The result is a hierarchy of inequalities for minors in terms of moments of the given state. The violation of any inequality is a sufficient condition for entanglement. Full entanglement can be certified for a manifold of multimode quantum states. A MATHEMATICA package is given for a systematic test of the hierarchy of conditions.

DOI
10.1103/physreva.74.030302

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Necessary and sufficient observable conditions for the non-negativity of all partial transpositions of multimode quantum states are derived. The result is a hierarchy of inequalities for minors in terms of moments of the given state. The violation of any inequality is a sufficient condition for entanglement. Full entanglement can be certified for a manifold of multimode quantum states. A MATHEMATICA package is given for a systematic test of the hierarchy of conditions.
We demonstrate how to efficiently derive a broad class of inequalities for entanglement detection in multimode continuous variable systems. The separability conditions are established from partial transposition (PT) in combination with several distinct necessary conditions for a quantum physical state, which include previously established inequalities as special cases. Remarkably, our method enables us to support Peres' conjecture to its full generality within the framework of Cavalcanti-Foster-Reid-Drummond multipartite Bell inequality [Phys. Rev. Lett. 99, 210405 (2007)] that the nonlocality necessarily implies negative PT entangled states.
We derive a hierarchy of continuous-variable multipartite entanglement conditions in terms of second-order moments of position and momentum operators that generalize existing criteria. Each condition corresponds to a convex optimization problem which, given the covariance matrix of the state, can be numerically solved in a straightforward way. The conditions are independent of partial transposition and thus are also able to detect bound entangled states. Our approach can be used to obtain an analytical condition for genuine multipartite entanglement. We demonstrate that even a special case of our conditions can detect entanglement very efficiently. Using multiobjective optimization, it is also possible to numerically verify genuine entanglement of some experimentally realizable states.
The entanglement produced by a bilinear Hamiltonian in continuous variables has been thoroughly studied and widely used. In contrast, the physics of entanglement resulting from nonlinear interaction described by partially degenerate high-order Hamiltonians remains unclear. Here, we derive a hierarchy of sufficient and necessary conditions for the positive-partial-transposition separability of bipartite nonlinear quantum states. The proposed criteria detect the nonpositive-partial-transposition inseparability of higher-order moments of states, which provides a systematic framework for the characterization of this kind of entanglement. Through numerical simulation of cubic and quartic Hamiltonians, we demonstrate the existence and competition of a hierarchy of entanglement witnesses, revealing the mechanism underlying such entanglement. Our results may provide a new direction in continuous variable quantum information processing.
A continuous-variable Bell inequality, valid for an arbitrary number of observers measuring observables with an arbitrary number of outcomes, was recently introduced [Cavalcanti, Phys. Rev. Lett. 99, 210405 (2007)10.1103/PhysRevLett.99.210405]. We prove that any n-mode quantum state violating this inequality with quadrature measurements necessarily has a negative partial transposition. Our results thus establish the first link between nonlocality and bound entanglement for continuous-variable systems.