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In the present paper we study the entanglement properties of thermal (a.k.a.
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The symplectic eigenvalues of a covariance matrix γ \gamma are simply the eigenvalues that are preserved under symplectic transformations and therefore condense the essential information of the covariance matrix, see for instance [ 18 ] or references therein. They can be calculated as the absolute values of the common eigenvalues of the matrix i σ γ i\sigma\gamma . Symplectic eigenvalues always appear with double multiplicity since the position and the momentum are treated interchangeably. This implies that for a single-mode covariance matrix ( 2 × 2 2\times 2 ), its determinant is just the square of the single symplectic eigenvalue
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L. Heaney, J. Anders, D. Kaszlikowski and V. Vedral (2007), Spatial Entanglement From Off-Diagonal Long Range Order in a BEC
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J. Anders (2008), Thermal state entanglement in harmonic lattices
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D. Cavalcanti, A. Ferraro, A. Garcia-Saez and A. Acin (2008), Thermal bound entanglement and area law
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