logo CRAS
Comptes Rendus. Mathématique
Théorie spectrale
On the disentanglement of Gaussian quantum states by symplectic rotations
[Sur la désintrication des états quantiques Gaussiens par des rotations symplectiques]
Comptes Rendus. Mathématique, Tome 358 (2020) no. 4, pp. 459-462.

Nous montrons que chaque état quantique Gaussien peut-être rendu séparable (= « désintriqué ») par conjugaison avec un opérateur unitaire associé via le groupe métaplectique à une rotation symplectique. Pour cela nous utilsons la condition de séparabilité de Werner et Wolf sur la matrice de covariance ainsi que la covariance symplectique des opérateurs pseudo-différentiels de Weyl.

We show that every Gaussian mixed quantum state can be disentangled by conjugation with a unitary operator corresponding to a symplectic rotation via the metaplectic representation of the symplectic group. The main tools we use are the Werner–Wolf condition for separability on covariance matrices and the symplectic covariance of Weyl pseudo-differential operators.

Reçu le : 2020-03-19
Révisé le : 2020-04-23
Accepté le : 2020-04-24
Publié le : 2020-07-28
DOI : https://doi.org/10.5802/crmath.57
@article{CRMATH_2020__358_4_459_0,
     author = {Maurice A. de Gosson},
     title = {On the disentanglement of Gaussian quantum states by symplectic rotations},
     journal = {Comptes Rendus. Math\'ematique},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {4},
     year = {2020},
     pages = {459-462},
     doi = {10.5802/crmath.57},
     language = {en},
     url = {comptes-rendus.academie-sciences.fr/mathematique/item/CRMATH_2020__358_4_459_0/}
}
Maurice A. de Gosson. On the disentanglement of Gaussian quantum states by symplectic rotations. Comptes Rendus. Mathématique, Tome 358 (2020) no. 4, pp. 459-462. doi : 10.5802/crmath.57. https://comptes-rendus.academie-sciences.fr/mathematique/item/CRMATH_2020__358_4_459_0/

[1] Gerardo Adesso; Fabrizio Illuminati Entanglement in continuous-variable systems: recent advances and current perspectives, J. Phys. A, Math. Theor., Volume 40 (2007) no. 28, pp. 7821-7880 | Zbl 1117.81009

[2] Gerardo Adesso; Sammy Ragy; Antony R. Lee Continuous variable quantum information: Gaussian states and beyond, Open Syst. Inf. Dyn., Volume 21 (2014) no. 1-2, 1440001, 1440001, 1440001, 1440001, 47 pages | Zbl 1295.81026

[3] Vladimir I. Arnol’d Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Springer, 1989

[4] Nuno Costa Dias; João Nuno Prata The Narcowich–Wigner spectrum of a pure state, Rep. Math. Phys., Volume 63 (2009) no. 1, pp. 43-54 | Zbl 1179.81105

[5] Maurice A. de Gosson Symplectic geometry and quantum mechanics, Operator Theory: Advances and Applications, Volume 166, Springer, 2006 | Zbl 1098.81004

[6] Maurice A. de Gosson The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?, Found. Phys., Volume 39 (2009) no. 2, pp. 194-214 | Zbl 1165.81030

[7] Maurice A. de Gosson Mixed quantum states with variable Planck constant, Phys. Lett., A, Volume 381 (2017) no. 36, pp. 3033-3037 | Zbl 1375.83075

[8] Ludovico Lami; Alessio Serafini; Gerardo Adesso Gaussian entanglement revisited, New J. Phys., Volume 20 (2018), 023030, 023030, 023030, 023030 | Article

[9] Robert G. Littlejohn The semiclassical evolution of wave packets, Phys. Rep., Volume 138 (1986) no. 4-5, pp. 193-291 | Article

[10] R. F. Werner; Michael M. Wolf Bound entangled Gaussian states, Phys. Rev. Lett., Volume 86 (2001) no. 16, pp. 3658-3661 | Article

[11] Michael M. Wolf; Jens Eisert; Martin B. Plenio Entangling power of passive optical elements, Phys. Rev. Lett., Volume 90 (2003) no. 4, 047904, 047904, 047904 | Article

[12] Michael M. Wolf; Geza Giedke; J. Ignacio Cirac Extremality of Gaussian quantum states, Phys. Rev. Lett., Volume 96 (2006) no. 8, 080502, 080502, 080502, 080502 | Article