On Orthogonal Projections of Symplectic Balls

Dias, Nuno C.; de Gosson, Maurice A.; Prata, João N.

Comptes Rendus. Mathématique, Tome 362 (2024) no. G3, pp. 217-227

We study the orthogonal projections of symplectic balls in ${ℝ}^{2n}$ on complex subspaces. In particular we show that these projections are themselves symplectic balls under a certain complexity assumption. Our main result is a refinement of a recent very interesting result of Abbondandolo and Matveyev extending the linear version of Gromov’s non-squeezing theorem. We use a conceptually simpler approach where the Schur complement of a matrix plays a central role. An application to the partial traces of density matrices is given.

Nous étudions les projections orthogonales de boules symplectiques dans ${ℝ}^{2n}$ sur des sous-espaces complexes. En particulier, nous montrons que ces projections sont elles-mêmes des boules symplectiques sous une certaine hypothèse de complexité. Notre résultat principal est une amélioration d’un résultat récent et très intéressant d’Abbondandolo et Matveyev, qui étend la version linéaire du théorème de non-plongement de Gromov. Nous utilisons une approche conceptuellement plus simple où le complément de Schur d’une matrice joue un rôle central. Une application aux traces partielles de matrices densité est donnée.

Keywords: Symplectic ball, orthogonal projection, Gromov’s non-squeezing theorem
Mots-clés : boule symplectique, projection orthogonale, théorème de non-plongement de Gromov

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