We study the orthogonal projections of symplectic balls in 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 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.
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Mots-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
CC-BY 4.0
@article{CRMATH_2024__362_G3_217_0,
author = {Nuno C. Dias and Maurice A. de Gosson and Jo\~ao N. Prata},
title = {On {Orthogonal} {Projections} of {Symplectic} {Balls}},
journal = {Comptes Rendus. Math\'ematique},
pages = {217--227},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.542},
language = {en},
}
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] On the local systolic optimality of Zoll contact forms, Geom. Funct. Anal., Volume 33 (2023), pp. 299-363 | DOI | MR | Zbl
[2] Sharp systolic inequalities for Reeb flows on the three-sphere, Invent. Math., Volume 211 (2018) no. 2, pp. 687-778 | DOI | MR | Zbl
[3] How large is the shadow of a symplectic ball?, J. Topol. Anal., Volume 5 (2013) no. 1, pp. 87-119 | DOI | MR
[4] Geometric measure theory, Grundlehren der Mathematischen Wissenschaften, 153, Springer, 1969
[5] Symplectic geometry and quantum mechanics, Operator Theory: Advances and Applications, 166, Springer, 2006 | DOI
[6] Symplectic methods in harmonic analysis and in mathematical physics, Pseudo-Differential Operators. Theory and Applications, 7, Springer, 2011
[7] 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] Pseudo holomorphic curves in symplectic manifolds, Invent. Math., Volume 82 (1985), pp. 307-347 | DOI | MR | Zbl
[9] Symplectic embeddings of polydisks, Invent. Math., Volume 172 (2008) no. 3, pp. 477-489 | DOI | MR
[10] Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts, Birkhäuser, 1994 | DOI
[11] Gaussian entanglement revisited, New J. Phys., Volume 20 (2018), 023030
[12] Symplectic embedding problems, old and new, Bull. Am. Math. Soc., Volume 55 (2018) no. 2, pp. 139-182 | DOI | MR
[13] Gaussian-Wigner distributions in quantum mechanics and optics, Phys. Rev. A, Volume 36 (1987) no. 8, 3868, pp. 3868-3880 | DOI | MR
[14] 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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