arXiv:2003.11049v2 [quant-ph] 26 Mar 2020 On the Disentanglement of Gaussian Quantum States by Symplectic Rotations Sur la D´ esintrication des ´ Etats Quantiques Gaussiens par des Rotations Symplectiques Maurice A. de Gosson ∗ Universit¨ at Wien Fakult¨ at f¨ ur Mathematik (NuHAG) Oskar-Morgenstern-Platz 1, 1090 Wien (AUSTRIA) March 27, 2020 Abstract We show that every Gaussian mixed quantum state can be disen- tangled by conjugation with a unitary operator. The main tools we use are the Werner–Wolf condition for separability on covariance matrices and the symplectic covariance of Weyl pseudo-differential operators. Abstract Nous montrons que chaque ´ etat quantique Gaussien peut-ˆ etre rendu s´ eparable (= “d´ esintriqu´ e”) par conjugaison avec un op´ erateur unitaire associ´ e` a une rotation symplectique. Pour cela nous utilsons la con- dition de s´ eparabilit´ e de Werner et Wolf sur la matrice de covariance ainsi que la covariance symplectique des op´ erateurs pseudo-diff´ erentiels de Weyl. Let R 2n = R 2n A ⊕ R 2n B be the phase space of a bipartite system (n A ≥ 1, n B ≥ 1). We will use the following phase space variable ordering: z = (z A ,z B )= z A ⊕ z B with z A =(x 1 ,p 1 , ..., x n A ,p n A ) and z B =(x n A +1 ,p n A +1 , ..., x n ,p n ). We equip the symplectic spaces R 2n A and R 2n B with their canon- ical bases. The symplectic structure on R 2n is then σ(z,z ′ )= Jz · z ′ with * maurice.de.gosson@univie.ac.at 1