Non linear Schrödinger limit of bosonic ground states, again
Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 69-91.

I review an information-theoretic variant of the quantum de Finetti theorem due to Brandão and Harrow and discuss its applications to the topic of bosonic mean-field limits. This leads to slightly improved methods for the derivation of the local non-linear Schrödinger energy functional from many-body quantum mechanics.

Reçu le : 2019-03-26
Révisé le : 2020-02-20
Accepté le : 2020-02-25
Publié le : 2020-09-25
DOI : https://doi.org/10.5802/cml.62
Classification : 81V70,  35Q40
Mots clés: Many-body quantum mechanics, mean-field limits, Schrödinger operators, de Finetti theorem, quantum information
@article{CML_2020__12_1_69_0,
     author = {Nicolas Rougerie},
     title = {Non linear Schr\"odinger limit of bosonic ground states, again},
     journal = {Confluentes Mathematici},
     pages = {69--91},
     publisher = {Institut Camille Jordan},
     volume = {12},
     number = {1},
     year = {2020},
     doi = {10.5802/cml.62},
     language = {en},
     url = {cml.centre-mersenne.org/item/CML_2020__12_1_69_0/}
}
Nicolas Rougerie. Non linear Schrödinger limit of bosonic ground states, again. Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 69-91. doi : 10.5802/cml.62. https://cml.centre-mersenne.org/item/CML_2020__12_1_69_0/

[1] Z. Ammari and F. Nier, Mean field limit for bosons and infinite dimensional phase-space analysis, Ann. Henri Poincaré, 9 (2008), pp. 1503–1574. | Article | MR 2465733 | Zbl 1171.81014

[2] C. Bardos, F. Golse, and N. J. Mauser, Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal., 7 (2000), pp. 275–293. | Article | Zbl 1003.81027

[3] N. Benedikter, G. de Oliveira, and B. Schlein, Quantitative Derivation of the Gross-Pitaevskii Equation, Comm. Pure App. Math., 68 (2015), pp. 1399–1482. | Article | MR 3366749 | Zbl 1320.35318

[4] C. Boccato, C. Brennecke, S. Cenatiempo, and B. Schlein, Optimal Rate for Bose-Einstein Condensation in the Gross-Pitaevskii Regime, Communications in Mathematical Physics, (2019). | Article | MR 4103969 | Zbl 1439.82004

[5] F. Brandão, M. Christandl, A. Harrow, and M. Walter, The Mathematics of Entanglement, 2016. arXiv:1604.01790

[6] F. Brandão and A. Harrow, Quantum de Finetti Theorems under Local Measurements with Applications, Commun. Math. Phys., 353 (2017), pp. 469–506. | Article | MR 3649478 | Zbl 1380.81068

[7] E. A. Carlen and E. H. Lieb, Remainder terms for some quantum entropy inequalities, J. Math. Phys., 55 (2014), p. 042201. | Article | MR 3390588 | Zbl 1292.81021

[8] X. Chen and J. Holmer, Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation, Arch. Rat. Mech. Anal., 221 (2016), pp. 631–676. | Article | Zbl 1338.35407

[9] —, The rigorous derivation of the 2D cubic focusing NLS from quantum many-body evolution, Int Math Res Notices, (2016). | Article

[10] G. Chiribella, On quantum estimation, quantum cloning and finite quantum de Finetti theorems, in Theory of Quantum Computation, Communication, and Cryptography, vol. 6519 of Lecture Notes in Computer Science, Springer, 2011. | Article | Zbl 1310.81034

[11] M. Christandl, R. König, G. Mitchison, and R. Renner, One-and-a-half quantum de Finetti theorems, Comm. Math. Phys., 273 (2007), pp. 473–498. | Article | MR 2318315 | Zbl 1126.81032

[12] L. Erdös, B. Schlein, and H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), pp. 515–614. | Article | Zbl 1123.35066

[13] L. Erdős, B. Schlein, and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc., 22 (2009), pp. 1099–1156. | Article | MR 2525781 | Zbl 1207.82031

[14] J. Fröhlich, A. Knowles, and S. Schwarz, On the mean-field limit of bosons with Coulomb two-body interaction, Commun. Math. Phys., 288 (2009), pp. 1023–1059. | Article | MR 2504864 | Zbl 1177.82016

[15] T. Girardot, Average field approximation for almost bosonic anyons in a magnetic field, J. Math. Physics 61, 071901 (2020). | Article | MR 4123638

[16] A. Harrow, The church of the symmetric subspace, 2013. arXiv:1308.6595

[17] M. Hayashi, Quantum information, Springer-Verlag, 2006. | Zbl 1195.81031

[18] R. L. Hudson and G. R. Moody, Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheor. und Verw. Gebiete, 33 (1975/76), pp. 343–351. | Article | MR 397421 | Zbl 0304.60001

[19] M. Jeblick, N. Leopold, and P. Pickl, Derivation of the time dependent gross-pitaevskii equation in two dimensions, Comm. Math. Physics, 372, 1–69 (2019). | Article | MR 4031794 | Zbl 07136897

[20] R. König and R. Renner, A de Finetti representation for finite symmetric quantum states, J. Math. Phys., 46 (2005), p. 122108. | Article | MR 2194019 | Zbl 1111.81020

[21] M. Lewin, Mean-Field limit of Bose systems: rigorous results, 2015. arXiv:1510.04407

[22] —, Geometric methods for nonlinear many-body quantum systems, J. Funct. Anal., 260 (2011), pp. 3535–3595. | Article | MR 2781970 | Zbl 1216.81180

[23] M. Lewin, P. Nam, and N. Rougerie, Derivation of Hartree’s theory for generic mean-field Bose systems, Adv. Math., 254 (2014), pp. 570–621. | Article | MR 3161107 | Zbl 1316.81095

[24] —, Remarks on the quantum de Finetti theorem for bosonic systems, Appl. Math. Res. Express (AMRX), 2015 (2015), pp. 48–63. | Article | Zbl 1310.81169

[25] —, The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases, Trans. Amer. Math. Soc, 368 (2016), pp. 6131–6157. | Article | Zbl 1392.35245

[26] —, A note on 2D focusing many-boson systems, Proc. Ame. Math. Soc., 145 (2017), pp. 2441–2454. | Article | MR 3626502 | Zbl 1366.35150

[27] —, Blow-up profile of rotating 2d focusing bose gases, in Macroscopic Limits of Quantum Systems, a conference in honor of Herbert Spohn’s 70th birthday, Springer, 2018, pp. 145–170. | Article | Zbl 1414.82032

[28] K. Li and G. Smith, Quantum de Finetti Theorems under fully-one-way adaptative measurements, Phys. Rev. Lett. 114, 114 (2015), p. 160503. | Article

[29] E. H. Lieb, Some convexity and subadditivity properties of entropy, Bulletin of the American Mathematical Society, 81 (1975), pp. 444–446. | Article | MR 356797 | Zbl 0308.94020

[30] E. H. Lieb and M. B. Ruskai, A fundamental property of quantum-mechanical entropy, Phys. Rev. Lett., 30 (1973), pp. 434–436. | Article | MR 373508

[31] —, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys., 14 (1973), pp. 1938–1941. With an appendix by B. Simon. | Article | MR 345558

[32] E. H. Lieb and R. Seiringer, Proof of Bose-Einstein Condensation for Dilute Trapped Gases, Phys. Rev. Lett., 88 (2002), p. 170409. | Article

[33] —, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Commun. Math. Phys., 264 (2006), pp. 505–537. | Article | MR 2215615 | Zbl 1233.82004

[34] E. H. Lieb, R. Seiringer, J. P. Solovej, and J. Yngvason, The mathematics of the Bose gas and its condensation, Oberwolfach Seminars, Birkhäuser, 2005. | Article | Zbl 1104.82012

[35] E. H. Lieb, R. Seiringer, and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A, 61 (2000), p. 043602. | Article

[36] —, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), pp. 17–31. | Article | MR 1868990 | Zbl 0996.82010

[37] E. H. Lieb and J. Yngvason, Ground state energy of the low density Bose gas, Phys. Rev. Lett., 80 (1998), pp. 2504–2507. | Article

[38] —, The ground state energy of a dilute two-dimensional Bose gas, J. Stat. Phys., 103 (2001), p. 509.

[39] D. Lundholm and N. Rougerie, The average field approximation for almost bosonic extended anyons, J. Stat. Phys., 161 (2015), pp. 1236–1267. | Article | MR 3422925 | Zbl 1334.81141

[40] A. Müller-Hermes and D. Reeb, Monotonicity of the quantum relative entropy under positive maps, Annales Henri Poincaré, 18 (2017), pp. 1777–1788. | Article | MR 3635967 | Zbl 1366.81045

[41] P. Nam and M. Napiórkowski, Norm approximation for many-body quantum dynamics: focusing case in low dimensions, Adv. Math. 350, 547–587 (2019). | Article | MR 3948169 | Zbl 1416.81059

[42] P. Nam and N. Rougerie, Improved stability for 2D attractive Bose gases, J. Math. Physics 61, 021901 (2020). | Article | MR 4068754 | Zbl 1439.81099

[43] P. T. Nam, N. Rougerie, and R. Seiringer, Ground states of large Bose systems: The Gross-Pitaevskii limit revisited, Analysis and PDEs, 9 (2016), pp. 459–485. | Article | MR 3513141 | Zbl 1342.35284

[44] M. Ohya and D. Petz, Quantum entropy and its use, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1993. | Article | Zbl 0891.94008

[45] P. Pickl, Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction, J. Stat. Phys., 140 (2010), pp. 76–89. | Article | MR 2651439 | Zbl 1202.82059

[46] —, A simple derivation of mean-field limits for quantum systems, Lett. Math. Phys., 97 (2011), pp. 151–164. | Article | MR 2821235 | Zbl 1242.81150

[47] I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Commun. Math. Phys., 291 (2009), pp. 31–61. | Article | MR 2530155 | Zbl 1186.82051

[48] N. Rougerie, De Finetti theorems, mean-field limits and Bose-Einstein condensation. LMU lecture notes 2014. arXiv:1506.05263

[49] —, Théorèmes de De Finetti, limites de champ moyen et condensation de Bose-Einstein, Les cours Peccot, Spartacus IDH, Paris, 2016. Cours Peccot, Collège de France : février-mars 2014.

[50] —, Scaling limits of bosonic ground states, from many-body to nonlinear Schrödinger , 2020. arXiv:2002.02678

[51] R. Schatten, Norm Ideals of Completely Continuous Operators, vol. 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge, 1960. | Article | Zbl 0090.09402

[52] R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Commun. Math. Phys., 229 (2002), pp. 491–509. | Article | Zbl 1004.82003

[53] —, Ground state asymptotics of a dilute, rotating gas, J. Phys. A, 36 (2003), pp. 9755–9778. | Article | MR 2007281 | Zbl 1151.82401

[54] B. Simon, Trace ideals and their applications, vol. 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979. | Zbl 0423.47001

[55] A. Triay, Derivation of the dipolar Gross–Pitaevskii energy, SIAM J. Math. Anal., 50 (2018), pp. 33–63. | Article | MR 3740324 | Zbl 1387.35514