Volume 3 (2020)
Ioos, Louis; Kaminker, Victoria; Polterovich, Leonid; Shmoish, Dor
Spectral aspects of the Berezin transform
Annales Henri Lebesgue, Volume 3 (2020), pp. 1343-1387.

Metadata

DOI10.5802/ahl.63
Keywords Berezin–Toeplitz quantization, Berezin transform, Laplace–Beltrami operator, balanced metric, Positive Operator Valued Measure

Abstract

We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin–Toeplitz quantization, Donaldson’s dynamical system on the space of Hermitian inner products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results confirm a prediction of Donaldson for the spectrum of the Q-operator on Kähler manifolds with constant scalar curvature, and yield exponential convergence of Donaldson’s iterations to the fixed point. Furthermore, viewing POVMs as data clouds, we study their spectral features via geometry of measure metric spaces and the diffusion distance.


References

[AS17] Aubrun, Guillaume; Szarek, Stanislaw J. Alice and Bob meet Banach. The interface of asymptotic geometric analysis and quantum information theory, Mathematical Surveys and Monographs, Volume 223, American Mathematical Society, 2017 | Zbl

[Ber75] Berezin, Feliks Alexandrovitch Quantization, Math. USSR, Izv., Volume 8 (1975) no. 5, pp. 1116-1175

[BGL13] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, Volume 348, Springer, 2013 | Zbl

[BGV92] Berline, Nicole; Getzler, Evra; Vergne, Michèle Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften, Volume 298, Springer, 1992 | MR | Zbl

[BLPY16] Busch, Paul; Lahti, Pekka J.; Pellonpää, Juha-Pekka; Ylinen, Kari Quantum measurement, Theoretical and Mathematical Physics, Springer, 2016 | Zbl

[BLY94] Bourguignon, Jean-Pierre; Li, Peter; Yau, Shing-Tung Upper bound for the first eigenvalue of algebraic submanifolds, Comment. Math. Helv., Volume 69 (1994) no. 2, pp. 199-207 | DOI | MR | Zbl

[BMS94] Bordemann, Martin; Meinrenken, Eckard; Schlichenmaier, Martin Toeplitz quantization of Kähler manifolds and gl (N), N→∞ limits, Commun. Math. Phys., Volume 165 (1994) no. 2, pp. 281-296 | DOI | Zbl

[CDS07] Chiribella, Giulio; D’Ariano, Giacomo Mauro; Schlingemann, Dirk How continuous quantum measurements in finite dimensions are actually discrete, Phys. Rev. Lett., Volume 98 (2007) no. 19, 190043, 4 pages | MR | Zbl

[CL06] Coifman, Ronald Raphaël; Lafon, Stéphane Diffusion maps, Appl. Comput. Harmon. Anal., Volume 21 (2006) no. 1, pp. 5-30 | DOI | MR | Zbl

[CP18a] Charles, Laurent; Polterovich, Leonid Quantum speed limit versus classical displacement energy, Ann. Henri Poincaré, Volume 19 (2018) no. 4, pp. 1215-1257 | DOI | MR | Zbl

[CP18b] Charles, Laurent; Polterovich, Leonid Sharp correspondence principle and quantum measurements, St. Petersbg. Math. J., Volume 29 (2018) no. 1, pp. 177-207 | DOI | MR

[CR12] Combescure, Monique; Robert, Didier Coherent states and applications in mathematical physics, Theoretical and Mathematical Physics, Springer, 2012 | DOI | Zbl

[Del] Deleporte, Alix private correspondence

[Dia88] Diaconis, Persi Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture Notes – Monograph Series, Volume 11, Institute of Mathematical Statistics, 1988 | MR | Zbl

[DLM06] Dai, Xianzhe; Liu, Kefeng; Ma, Xiaonan On the asymptotic expansion of Bergman kernel, J. Differ. Geom., Volume 72 (2006) no. 1, pp. 1-41 | MR | Zbl

[Don09] Donaldson, Simon Kirwan Some numerical results in complex differential geometry, Pure Appl. Math. Q., Volume 5 (2009) no. 2, pp. 571-618 (Special Issue: In honor of Friedrich Hirzebruch. Part 1) | DOI | MR

[EF02] Eldar, Yonina Chana; Forney, G. David Jun. Optimal tight frames and quantum measurement, IEEE Trans. Inf. Theory, Volume 48 (2002) no. 3, pp. 599-610 | DOI | MR | Zbl

[Eng00] Engliš, Miroslav A Forelli–Rudin construction and asymptotics of weighted Bergman kernels, J. Funct. Anal., Volume 177 (2000) no. 2, pp. 257-281 | DOI | MR | Zbl

[Far99] Farebrother, Richard William Fitting Linear Relationships: A History of the Calculus of Observations 1750–1900, Springer Series in Statistics, Springer, 1999 | Zbl

[Fin10] Fine, Joel Calabi flow and projective embeddings, J. Differ. Geom., Volume 84 (2010) no. 3, pp. 489-523 (with an appendix by Kefeng Liu and Xiaonan Ma) | DOI | MR | Zbl

[Fin12a] Fine, Joel Notes on canonical Kähler metrics and quantisation, 2012 (preprint, available at http://homepages.vub.ac.be/~joelfine/preprints/2012-07_Cologne.pdf)

[Fin12b] Fine, Joel Quantization and the Hessian of Mabuchi energy, Duke Math. J., Volume 161 (2012) no. 14, pp. 2753-2798 | DOI | MR | Zbl

[FM05] Flaschka, Hermann; Millson, John Bending flows for sums of rank one matrices, Can. J. Math., Volume 57 (2005) no. 1, pp. 114-158 | DOI | MR | Zbl

[GM00] Giannopoulos, Apostolos A.; Milman, Vitali D. Extremal problems and isotropic positions of convex bodies, Isr. J. Math., Volume 117 (2000), pp. 29-60 | DOI | MR | Zbl

[GU88] Guillemin, Victor W.; Uribe, Alejandro The Laplace operator on the n-th tensor power of a line bundle: eigenvalues which are uniformly bounded in n, Asymptotic Anal., Volume 1 (1988) no. 2, pp. 105-113 | DOI | MR | Zbl

[Hay06] Hayashi, Masahito Quantum Information. An introduction, Springer, 2006 | Zbl

[Her70] Hersch, Joseph Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci., Paris, Sér. A, Volume 270 (1970), pp. 1645-1648 | Zbl

[ILMM17] Ioos, Louis; Lu, Wen; Ma, Xiaonan; Marinescu, George Berezin–Toeplitz quantization for eigenstates of the Bochner–Laplacian on symplectic manifolds, J. Geom. Anal., Volume 30 (2017) no. 3, pp. 2615-2646 | DOI | MR | Zbl

[Isa06] Isaacs, I. Martin Character Theory of Finite Groups, AMS Chelsea Publishing, 2006 (corrected reprint of the 1976 original) | Zbl

[Kan77] Kannai, Yakar Off diagonal short time asymptotics for fundamental solution of diffusion equation, Commun. Partial Differ. Equations, Volume 2 (1977) no. 8, pp. 781-830 | DOI | MR

[Kel09] Keller, Julien Ricci iterations on Kähler classes, J. Inst. Math. Jussieu, Volume 8 (2009) no. 4, pp. 743-768 | DOI | MR | Zbl

[KMS16] Keller, Julien; Meyer, Julien; Seyyedali, Reza Quantization of the Laplacian operator on vector bundles. I, Math. Ann., Volume 366 (2016) no. 3-4, pp. 865-907 | DOI | MR | Zbl

[KS01] Karabegov, Alexander V.; Schlichenmaier, Martin Identification of Berezin–Toeplitz deformation quantization, J. Reine Angew. Math., Volume 540 (2001), pp. 49-76 | MR | Zbl

[Lan98] Landsman, Nicolaas P. Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics, Springer, 1998

[LF18] Le Floch, Yohann A Brief Introduction to Berezin–Toeplitz Operators on Compact Kähler Manifolds, CRM Short Courses, Springer, 2018 | DOI | MR | Zbl

[LM07] Liu, Kefeng; Ma, Xiaonan A remark on: “Some numerical results in complex differential geometry”, Math. Res. Lett., Volume 14 (2007) no. 2, pp. 165-171 | Zbl

[LM10] Lebeau, Gilles; Michel, Laurent Semi-classical analysis of a random walk on a manifold, Ann. Probab., Volume 38 (2010) no. 1, pp. 277-315 | DOI | MR | Zbl

[LMM17] Lu, Wen; Ma, Xiaonan; Marinescu, George Donaldson’s Q-operators for symplectic manifolds, Sci. China, Math., Volume 60 (2017) no. 6, pp. 1047-1056 | MR | Zbl

[MM07] Ma, Xiaonan; Marinescu, George Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics, Volume 254, Birkhäuser, 2007 | MR | Zbl

[MM08] Ma, Xiaonan; Marinescu, George Generalized Bergman kernels on symplectic manifolds, Adv. Math., Volume 217 (2008) no. 4, pp. 1756-1815 | MR | Zbl

[MM12] Ma, Xiaonan; Marinescu, George Berezin–Toeplitz quantization on Kähler manifolds, J. Reine Angew. Math., Volume 662 (2012), pp. 1-56 | Zbl

[NC00] Nielsen, Michael A.; Chuang, Isaac L. Quantum Computation and Quantum Information, Cambridge University Press, 2000 | Zbl

[OC09] Oreshkov, Ognyan; Calsamiglia, John Distinguishability measures between ensembles of quantum states, Phys. Rev. A, Volume 79 (2009) no. 3, 032336

[RSVN18] Rouby, Ophélie; Sjoestrand, Johannes; Vu Ngoc, San Analytic Bergman operators in the semiclassical limit (2018) (https://arxiv.org/abs/1808.00199)

[Rud13] Rudolf, Daniel An introduction to Markov chain Monte Carlo on general state spaces, 2013 (preprint, available at http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.461.7352&rep=rep1&type=pdf)

[Sch10] Schlichenmaier, Martin Berezin–Toeplitz quantization for compact Kähler manifolds. A review of results, Adv. Math. Phys., Volume 2010 (2010), 927280, 38 pages | Zbl

[Wal18] Waldron, Shayne F. D. An Introduction to Finite Tight Frames, Applied and Numerical Harmonic Analysis, Birkhäuser; Springer, 2018 | MR | Zbl

[Zha98] Zhang, Genkai Berezin transform on compact Hermitian symmetric spaces, Manuscr. Math., Volume 97 (1998) no. 3, pp. 371-388 | DOI | MR | Zbl