Comptes Rendus
Article de recherche - Géométrie et Topologie, Physique mathématique
On Orthogonal Projections of Symplectic Balls
[Sur les projections orthogonales de boules symplectiques]
Nuno C. Dias1, 2  ; Maurice A. de Gosson3  ; João N. Prata1, 2
1 Grupo de Física Matemática, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2 Escola Superior Náutica Infante D. Henrique. Av. Eng. Bonneville Franco, 2770-058 Paço d’Arcos, Portugal
3 University of Vienna, Faculty of Mathematics (NuHAG), Vienna, Austria
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 217-227.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.542
Keywords: Symplectic ball, orthogonal projection, Gromov’s non-squeezing theorem
Mot clés : boule symplectique, projection orthogonale, théorème de non-plongement de Gromov
Nuno C. Dias 1, 2 ; Maurice A. de Gosson 3 ; João N. Prata 1, 2

1 Grupo de Física Matemática, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2 Escola Superior Náutica Infante D. Henrique. Av. Eng. Bonneville Franco, 2770-058 Paço d’Arcos, Portugal
3 University of Vienna, Faculty of Mathematics (NuHAG), Vienna, Austria
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {On {Orthogonal} {Projections} of {Symplectic} {Balls}},
     journal = {Comptes Rendus. Math\'ematique},
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Nuno C. Dias; Maurice A. de Gosson; João N. Prata. On Orthogonal Projections of Symplectic Balls. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 217-227. doi : 10.5802/crmath.542. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.542/

[1] Alberto Abbondandolo; Gabriele Benedetti On the local systolic optimality of Zoll contact forms, Geom. Funct. Anal., Volume 33 (2023), pp. 299-363 | DOI | MR | Zbl

[2] Alberto Abbondandolo; Barney Bramham; Umberto L. Hryniewicz; Pedro A. S. Salomão Sharp systolic inequalities for Reeb flows on the three-sphere, Invent. Math., Volume 211 (2018) no. 2, pp. 687-778 | DOI | MR | Zbl

[3] Alberto Abbondandolo; Rostislav Matveyev How large is the shadow of a symplectic ball?, J. Topol. Anal., Volume 5 (2013) no. 1, pp. 87-119 | DOI | MR

[4] Herbert Federer Geometric measure theory, Grundlehren der Mathematischen Wissenschaften, 153, Springer, 1969

[5] Maurice de Gosson Symplectic geometry and quantum mechanics, Operator Theory: Advances and Applications, 166, Springer, 2006 | DOI

[6] Maurice de Gosson Symplectic methods in harmonic analysis and in mathematical physics, Pseudo-Differential Operators. Theory and Applications, 7, Springer, 2011

[7] Maurice de Gosson On Density Operators with Gaussian Weyl Symbols, Proceedings to the Conference MLTFA18 – Microlocal and Time-Frequency Analysis (Applied and Numerical Harmonic Analysis), Birkhäuser, 2018, pp. 191-206 | Zbl

[8] Mikhael Gromov Pseudo holomorphic curves in symplectic manifolds, Invent. Math., Volume 82 (1985), pp. 307-347 | DOI | MR | Zbl

[9] Larry Guth Symplectic embeddings of polydisks, Invent. Math., Volume 172 (2008) no. 3, pp. 477-489 | DOI | MR

[10] Helmut Hofer; Eduard Zehnder Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts, Birkhäuser, 1994 | DOI

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

[12] Felix Schlenk Symplectic embedding problems, old and new, Bull. Am. Math. Soc., Volume 55 (2018) no. 2, pp. 139-182 | DOI | MR

[13] R. Simon; E. C. G. Sudarshan; N. Mukunda Gaussian-Wigner distributions in quantum mechanics and optics, Phys. Rev. A, Volume 36 (1987) no. 8, 3868, pp. 3868-3880 | DOI | MR

[14] R. F. Werner; M. M. Wolf Bound Entangled Gaussian States, Phys. Rev. Lett., Volume 86 (2001) no. 16, 3658, pp. 3658-3661 | DOI

[15] The Schur Complement and its Applications (Fuzhen Zhang, ed.), Numerical Methods and Algorithms, 4, Springer, 2005 | DOI

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