Optimal estimates of trace distance between bosonic Gaussian states and applications to learning

Lennart Bittel; Francesco Anna Mele; Antonio Anna Mele; Salvatore Tirone; Ludovico Lami

doi:10.22331/q-2025-06-12-1769

Optimal estimates of trace distance between bosonic Gaussian states and applications to learning

Lennart Bittel¹, Francesco Anna Mele², Antonio Anna Mele¹, Salvatore Tirone^3,4, and Ludovico Lami^5,3,4,6

¹Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
²NEST, Scuola Normale Superiore and Istituto Nanoscienze, Piazza dei Cavalieri 7, IT-56126 Pisa, Italy
³QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands
⁴Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, the Netherlands
⁵Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
⁶Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands

Published:	2025-06-12, volume 9, page 1769
Editor:	Daniel Malz
Eprint:	arXiv:2411.02368v4
Doi:	https://doi.org/10.22331/q-2025-06-12-1769
Citation:	Quantum 9, 1769 (2025).

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Abstract

Gaussian states of bosonic quantum systems enjoy numerous technological applications and are ubiquitous in nature. Their significance lies in their simplicity, which in turn rests on the fact that they are uniquely determined by two experimentally accessible quantities, their first and second moments. But what if these moments are only known approximately, as is inevitable in any realistic experiment? What is the resulting error on the Gaussian state itself, as measured by the most operationally meaningful metric for distinguishing quantum states, namely, the trace distance? In this work, we fully resolve this question by demonstrating that if the first and second moments are known up to an error $\varepsilon$, the trace distance error on the state also scales as $\varepsilon$, and this functional dependence is optimal. To prove this, we establish tight bounds on the trace distance between two Gaussian states in terms of the norm distance of their first and second moments. As an application, we improve existing bounds on the sample complexity of tomography of Gaussian states.

Featured image: Trace distance upper bound between two Gaussian states in terms of their covariance matrices' distance.

Popular summary

How precise must experiments be to reconstruct a quantum state of light?

Gaussian states—such as coherent and squeezed states—are among the most important states in quantum optics and continuous-variable quantum information. Their widespread use comes from a remarkable feature: a Gaussian state is completely determined by just two quantities—its first moments (like average position and momentum) and second moments (like variances and covariances). These are quantities that can, in principle, be measured directly in the lab.

But no real experiment can measure with perfect accuracy. So what happens when these moments are only known approximately—say, with error at most $\varepsilon$? How close is the corresponding Gaussian state to the true one?

In this work, we provide a complete and optimal answer to this fundamental question. We prove that if the first and second moments are known up to error $\varepsilon$, then the error on the full quantum state—measured using the trace distance, which quantifies how well two quantum states can be distinguished—also scales linearly with $\varepsilon$. This scaling is not just a bound: it is optimal. No better general relationship is possible.
As a key application, we sharpen existing bounds on the sample complexity of Gaussian state tomography—that is, the number of measurements required to reconstruct an unknown Gaussian state from experimental data.

To reach this result, we develop sharp bounds linking trace distance and moment errors, and introduce a new tool—the derivative of a Gaussian state—which sheds light on the differential geometry of the Gaussian state manifold.

► BibTeX data

@article{Bittel2025optimalestimatesof,
  doi = {10.22331/q-2025-06-12-1769},
  url = {https://doi.org/10.22331/q-2025-06-12-1769},
  title = {Optimal estimates of trace distance between bosonic {G}aussian states and applications to learning},
  author = {Bittel, Lennart and Mele, Francesco Anna and Mele, Antonio Anna and Tirone, Salvatore and Lami, Ludovico},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1769},
  month = jun,
  year = {2025}
}

► References

[1] A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed., Texts and Monographs in Theoretical Physics (De Gruyter, Berlin, Germany, 2019).
https://doi.org/10.1515/9783110273403

[2] A. Serafini, Quantum continuous variables: A primer of theoretical methods (CRC Press, Taylor & Francis Group, Boca Raton, USA, 2017).
https://doi.org/10.1201/9781315118727

[3] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).
https://doi.org/10.1103/RevModPhys.84.621

[4] N. J. Cerf, G. Leuchs, and E. S. Polzik, Quantum information with continuous variables of atoms and light (Imperial College Press, London, UK, 2007).
http://emu.tind.io/record/298574

[5] G. Adesso, S. Ragy, and A. R. Lee, Open Systems amp; Information Dynamics 21, 1440001 (2014).
https://doi.org/10.1142/s1230161214400010

[6] S. Lloyd and S. L. Braunstein, Phys. Rev. Lett. 82, 1784 (1999).
https://doi.org/10.1103/PhysRevLett.82.1784

[7] D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001).
https://doi.org/10.1103/PhysRevA.64.012310

[8] N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, Phys. Rev. Lett. 97, 110501 (2006).
https://doi.org/10.1103/PhysRevLett.97.110501

[9] M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, New J. Phys. 16, 045014 (2014).
https://doi.org/10.1088/1367-2630/16/4/045014

[10] N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. Girvin, L. Jiang, M. Mirrahimi, M. Devoret, and R. Schoelkopf, Nature 536 (2016).
https://doi.org/10.1038/nature18949

[11] M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, Phys. Rev. X 6, 031006 (2016).
https://doi.org/10.1103/PhysRevX.6.031006

[12] J. Guillaud and M. Mirrahimi, Phys. Rev. X 9, 041053 (2019).
https://doi.org/10.1103/physrevx.9.041053

[13] K. Alexander, A. Bahgat, A. Benyamini, D. Black, D. Bonneau, S. Burgos, B. Burridge, G. Campbell, G. Catalano, A. Ceballos, C.-M. Chang, C. Chung, F. Danesh, T. Dauer, M. Davis, E. Dudley, P. Er-Xuan, J. Fargas, A. Farsi, C. Fenrich, J. Frazer, M. Fukami, Y. Ganesan, G. Gibson, M. Gimeno-Segovia, S. Goeldi, P. Goley, R. Haislmaier, S. Halimi, P. Hansen, S. Hardy, J. Horng, M. House, H. Hu, M. Jadidi, H. Johansson, T. Jones, V. Kamineni, N. Kelez, R. Koustuban, G. Kovall, P. Krogen, N. Kumar, Y. Liang, N. LiCausi, D. Llewellyn, K. Lokovic, M. Lovelady, V. Manfrinato, A. Melnichuk, M. Souza, G. Mendoza, B. Moores, S. Mukherjee, J. Munns, F.-X. Musalem, F. Najafi, J. L. O'Brien, J. E. Ortmann, S. Pai, B. Park, H.-T. Peng, N. Penthorn, B. Peterson, M. Poush, G. J. Pryde, T. Ramprasad, G. Ray, A. Rodriguez, B. Roxworthy, T. Rudolph, D. J. Saunders, P. Shadbolt, D. Shah, H. Shin, J. Smith, B. Sohn, Y.-I. Sohn, G. Son, C. Sparrow, M. Staffaroni, C. Stavrakas, V. Sukumaran, D. Tamborini, M. G. Thompson, K. Tran, M. Triplet, M. Tung, A. Vert, M. D. Vidrighin, I. Vorobeichik, P. Weigel, M. Wingert, J. Wooding, and X. Zhou, A manufacturable platform for photonic quantum computing (2024), arXiv:2404.17570.
arXiv:2404.17570

[14] L. Brenner, L. Caha, X. Coiteux-Roy, and R. Koenig, Factoring an integer with three oscillators and a qubit (2024), arXiv:2412.13164 [quant-ph].
arXiv:2412.13164

[15] M. M. Wolf, D. Pérez-García, and G. Giedke, Phys. Rev. Lett. 98, 130501 (2007).
https://doi.org/10.1103/PhysRevLett.98.130501

[16] M. Takeoka, S. Guha, and M. M. Wilde, Nature Comm. 5, 5235 (2014).
https://doi.org/10.1038/ncomms6235

[17] S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, Nature Comm. 8, 15043 (2017).
https://doi.org/10.1038/ncomms15043

[18] F. A. Mele, L. Lami, and V. Giovannetti, Phys. Rev. Lett. 129, 180501 (2022a).
https://doi.org/10.1103/PhysRevLett.129.180501

[19] F. A. Mele, L. Lami, and V. Giovannetti, Maximum tolerable excess noise in CV-QKD and improved lower bound on two-way capacities (2023a), arXiv:2303.12867.
https://doi.org/10.1038/s41566-024-01595-9
arXiv:2303.12867

[20] F. A. Mele, F. Salek, V. Giovannetti, and L. Lami, Quantum communication on the bosonic loss-dephasing channel (2024a), arXiv:2401.15634.
https://doi.org/10.1103/PhysRevA.110.012460
arXiv:2401.15634

[21] F. A. Mele, L. Lami, and V. Giovannetti, Phys. Rev. A 106, 042437 (2022b).
https://doi.org/10.1103/PhysRevA.106.042437

[22] F. A. Mele, G. D. Palma, M. Fanizza, V. Giovannetti, and L. Lami, Optical fibres with memory effects and their quantum communication capacities (2023b), arXiv:2309.17066 [quant-ph].
https://doi.org/10.1109/TIT.2024.3450501
arXiv:2309.17066

[23] T. L. S. Collaboration, Nature Phot. 7, 613 (2013).
https://doi.org/10.1038/nphoton.2013.177

[24] J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, Phys. Rev. Lett. 121, 160502 (2018).
https://doi.org/10.1103/PhysRevLett.121.160502

[25] V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. Lett. 86, 5870 (2001).
https://doi.org/10.1103/PhysRevLett.86.5870

[26] K. McCormick, J. Keller, S. Burd, D. Wineland, A. Wilson, and D. Leibfried, Nature 572, 1 (2019).
https://doi.org/10.1038/s41586-019-1421-y

[27] M. Ohliger, V. Nesme, D. Gross, Y.-K. Liu, and J. Eisert, Continuous variable quantum compressed sensing (2011), arXiv:1111.0853.
arXiv:1111.0853

[28] L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. Nam, V. D. Vaidya, M. Menotti, I. Dhand, Z. Vernon, N. Quesada, and J. Lavoie, Nature 606, 75 (2022).
https://doi.org/10.1038/s41586-022-04725-x

[29] H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, Science 370, 1460–1463 (2020).
https://doi.org/10.1126/science.abe8770

[30] D. Hangleiter and J. Eisert, Rev. Mod. Phys. 95, 035001 (2023).
https://doi.org/10.1103/RevModPhys.95.035001

[31] F. H. B. Somhorst, R. van der Meer, M. C. Anguita, R. Schadow, H. J. Snijders, M. de Goede, B. Kassenberg, P. Venderbosch, C. Taballione, J. P. Epping, H. H. van den Vlekkert, J. F. F. Bulmer, J. Lugani, I. A. Walmsley, P. W. H. Pinkse, J. Eisert, N. Walk, and J. J. Renema, Nature Comm. 14, 3895 (2023).
https://doi.org/10.1038/s41467-023-38413-9

[32] C. W. Helstrom, Quantum detection and estimation theory (Academic press, New York, USA, 1976).
https://doi.org/10.1007/BF01007479

[33] A. S. Holevo, Trudy Mat. Inst. Steklov 124, 3 (1976), (English translation: Proc. Steklov Inst. Math. 124:1–140, 1978).
https://doi.org/10.1070/RM9856

[34] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, 2010).
https://doi.org/10.1017/CBO9780511976667

[35] M. M. Wilde, Quantum Information Theory, 2nd ed. (Cambridge University Press, 2017).
https://doi.org/10.1017/CBO9781139525343

[36] S. Khatri and M. M. Wilde, Principles of quantum communication theory: A modern approach (2020).
https://doi.org/10.48550/ARXIV.2011.04672

[37] J. Watrous, The theory of quantum information (Cambridge University Press, 2018).
https://doi.org/10.1017/9781316848142

[38] A. S. Holevo, On estimates of trace-norm distance between quantum Gaussian states (2024a), arXiv:2408.11400 [quant-ph].
arXiv:2408.11400

[39] A. S. Holevo, On estimates of the bures distance between bosonic gaussian states (2024b), arXiv:2412.04875 [quant-ph].
arXiv:2412.04875

[40] F. A. Mele, A. A. Mele, L. Bittel, J. Eisert, V. Giovannetti, L. Lami, L. Leone, and S. F. E. Oliviero, Learning quantum states of continuous variable systems (2024b), arXiv:2405.01431 [quant-ph].
arXiv:2405.01431

[41] L. Bittel, F. A. Mele, A. A. Mele, S. Tirone, and L. Lami, Optimal estimates of trace distance between bosonic gaussian states and applications to learning (2024a), arXiv:2411.02368 [quant-ph].
arXiv:2411.02368

[42] J. Haah, R. Kothari, R. O’Donnell, and E. Tang, in 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2023) p. 363–390.
https://doi.org/10.1109/focs57990.2023.00028

[43] L. Banchi, S. Braunstein, and S. Pirandola, Phys. Rev. Lett. 115, 260501 (2015).
https://doi.org/10.1103/physrevlett.115.260501

[44] A. Anshu and S. Arunachalam, Nat. Rev. Phys. 6, 59 (2023).
https://doi.org/10.1038/s42254-023-00662-4

[45] L. Aolita, C. Gogolin, M. Kliesch, and J. Eisert, Nat Commun 6, 8498 (2015).
https://doi.org/10.1038/ncomms9498

[46] S. Gandhari, V. V. Albert, T. Gerrits, J. M. Taylor, and M. J. Gullans, Precision bounds on continuous-variable state tomography using classical shadows (2023), arxiv:2211.05149.
arXiv:2211.05149

[47] S. Becker, N. Datta, L. Lami, and C. Rouzé, IEEE Trans. Inf. Theory 70, 3427 (2024).
https://doi.org/10.1109/TIT.2024.3357972

[48] C. Oh, S. Chen, Y. Wong, S. Zhou, H.-Y. Huang, J. A. H. Nielsen, Z.-H. Liu, J. S. Neergaard-Nielsen, U. L. A., L. Jiang, and J. Preskill, Entanglement-enabled advantage for learning a bosonic random displacement channel (2024), arXiv:2402.18809.
https://doi.org/10.1103/PhysRevLett.133.230604
arXiv:2402.18809

[49] O. Fawzi, A. Oufkir, and R. Salzmann, Optimal fidelity estimation from binary measurements for discrete and continuous variable systems (2024), arXiv:2409.04189 [quant-ph].
arXiv:2409.04189

[50] T. Möbus, A. Bluhm, M. C. Caro, A. H. Werner, and C. Rouzé, Dissipation-enabled bosonic hamiltonian learning via new information-propagation bounds (2023), arXiv:2307.15026 [quant-ph].
arXiv:2307.15026

[51] V. Upreti and U. Chabaud, An efficient quantum state verification framework and its application to bosonic systems (2024), arXiv:2411.04688 [quant-ph].
arXiv:2411.04688

[52] M. Fanizza, C. Rouzé, and D. S. França, Efficient hamiltonian, structure and trace distance learning of gaussian states (2024), arXiv:2411.03163 [quant-ph].
arXiv:2411.03163

[53] R. Nichols, P. Liuzzo-Scorpo, P. A. Knott, and G. Adesso, Phys. Rev. A 98, 012114 (2018).
https://doi.org/10.1103/PhysRevA.98.012114

[54] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006).
https://doi.org/10.1103/PhysRevLett.96.010401

[55] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, 222–229 (2011).
https://doi.org/10.1038/nphoton.2011.35

[56] A. Monras, Phase space formalism for quantum estimation of gaussian states (2013), arXiv:1303.3682 [quant-ph].
arXiv:1303.3682

[57] Y. Gao and H. Lee, The European Physical Journal D 68, 10.1140/epjd/e2014-50560-1 (2014).
https://doi.org/10.1140/epjd/e2014-50560-1

[58] A. Monras and F. Illuminati, Physical Review A 81, 10.1103/physreva.81.062326 (2010).
https://doi.org/10.1103/physreva.81.062326

[59] L. Bittel, A. A. Mele, J. Eisert, and L. Leone, arXiv (2024b), arXiv:2409.17953 [quant-ph].
arXiv:2409.17953

[60] Z. Huang and M. M. Wilde, Information geometry of bosonic gaussian thermal states (2024a), arXiv:2411.18268 [quant-ph].
arXiv:2411.18268

[61] See the Supplemental Material for details on the derivation of the derivative of a Gaussian state, the proof of the main results, and the applications of our results to quantum state tomography.

[62] S. S. Barsov and V. V. Ul'yanov, Soviet Mathematics, Doklady 34, 462 (1987), available at https://www.researchgate.net/publication/268861237_Estimates_of_the_proximity_of_Gaussian_measures.
https://www.researchgate.net/publication/268861237_Estimates_of_the_proximity_of_Gaussian_measures

[63] L. Devroye, A. Mehrabian, and T. Reddad, The total variation distance between high-dimensional Gaussians with the same mean (2023), arXiv:1810.08693.
arXiv:1810.08693

[64] J. Arbas, H. Ashtiani, and C. Liaw, Polynomial time and private learning of unbounded Gaussian mixture models (2023), arXiv:2303.04288.
arXiv:2303.04288

[65] R. O'Donnell and J. Wright, Efficient quantum tomography (2015), arXiv:1508.01907.
arXiv:1508.01907

[66] J. Haah, A. Harrow, J. Zhengfeng, X. Wu, and N. Yu, IEEE Trans. Inf. Th. 10.1109/tit.2017.2719044 (2017).
https://doi.org/10.1109/tit.2017.2719044

[67] R. Kueng, H. Rauhut, and U. Terstiege, Low rank matrix recovery from rank one measurements (2014), arXiv:1410.6913.
arXiv:1410.6913

[68] S. Chen, B. Huang, J. Li, A. Liu, and M. Sellke, When does adaptivity help for quantum state learning? (2023), arXiv:2206.05265.
arXiv:2206.05265

[69] M. Guta, J. Kahn, R. Kueng, and J. Tropp, Fast state tomography with optimal error bounds (2018), arXiv:1809.11162.
arXiv:1809.11162

[70] M. Fanizza, N. Galke, J. Lumbreras, C. Rouzé, and A. Winter, Learning finitely correlated states: stability of the spectral reconstruction (2023), arXiv:2312.07516.
arXiv:2312.07516

[71] H. Huang, Y. Liu, M. Broughton, I. Kim, A. Anshu, Z. Landau, and J. R. McClean, in Proc. 56th Annu. ACM Symp. Theory Comput. (Association for Computing Machinery, New York, NY, USA, 2024) pp. 1343–1351.
https://doi.org/10.1145/3618260.3649722

[72] S. Arunachalam, S. Bravyi, A. Dutt, and T. J. Yoder, Optimal algorithms for learning quantum phase states (2023), arXiv:2208.07851.
arXiv:2208.07851

[73] A. Montanaro, Learning stabilizer states by bell sampling (2017), arXiv:1707.04012 [quant-ph].
arXiv:1707.04012

[74] S. Grewal, V. Iyer, W. Kretschmer, and D. Liang, Efficient learning of quantum states prepared with few non-Clifford gates (2023), arXiv:2305.13409.
arXiv:2305.13409

[75] L. Leone, S. Oliviero, and A. Hamma, Quantum 8, 1361 (2024).
https://doi.org/10.22331/q-2024-05-27-1361

[76] D. Hangleiter and M. Gullans, Bell sampling from quantum circuits (2024), arXiv:2306.00083.
https://doi.org/10.1103/PhysRevLett.133.020601
arXiv:2306.00083

[77] S. Aaronson and S. Grewal, Efficient tomography of non-interacting fermion states (2023), arXiv:2102.10458.
arXiv:2102.10458

[78] A. A. Mele and Y. Herasymenko, PRX Quantum 6, 10.1103/prxquantum.6.010319 (2025).
https://doi.org/10.1103/prxquantum.6.010319

[79] Z. Huang and M. M. Wilde, Information geometry of bosonic gaussian thermal states (2024b), arXiv:2411.18268 [quant-ph].
arXiv:2411.18268

[80] M. M. Wilde, M. Tomamichel, S. Lloyd, and M. Berta, Physical Review Letters 119, 10.1103/physrevlett.119.120501 (2017).
https://doi.org/10.1103/physrevlett.119.120501

[81] H. J. Groenewold, Physica 12, 405 (1946).
https://doi.org/10.1016/S0031-8914(46)80059-4

[82] L. Lami, B. Regula, X. Wang, R. Nichols, A. Winter, and G. Adesso, Phys. Rev. A 98, 022335 (2018).
https://doi.org/10.1103/PhysRevA.98.022335

[83] M. Keyl, J. Kiukas, and R. F. Werner, Rev. Math. Phys. 28, 1630001 (2016).
https://doi.org/10.1142/S0129055X16300016

[84] L. Narici and E. Beckenstein, Topological Vector Spaces (Chapman and Hall/CRC, 2010).
https://doi.org/10.1201/9781584888673

[85] L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis) (Springer-Verlag, 1990).
https://doi.org/10.1007/978-3-642-61497-2

[86] E. Zeidler, Applied Functional Analysis (Springer, New York, 1995).
https://doi.org/10.1007/978-1-4612-0821-1

[87] R. Bhatia, Positive definite matrices (Princeton university press, 2009).
https://doi.org/10.1515/9781400827787

[88] R. Bhatia and T. Jain, Journal of Mathematical Physics 56, 10.1063/1.4935852 (2015).
https://doi.org/10.1063/1.4935852

[89] G. Lugosi and S. Mendelson, Sub-gaussian estimators of the mean of a random vector (2017), arXiv:1702.00482 [math.ST].
arXiv:1702.00482

[90] R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2018).
https://doi.org/10.1017/9781108231596

[91] M. J. Wainwright, High-Dimensional Statistics: A Non-Asymptotic Viewpoint, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2019).
https://doi.org/10.1017/9781108627771

[92] R. Vershynin, Introduction to the non-asymptotic analysis of random matrices (2011), arXiv:1011.3027 [math.PR].
arXiv:1011.3027

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[4] Xiaobin Zhao, Pengcheng Liao, Francesco Anna Mele, Ulysse Chabaud, and Quntao Zhuang, "Complexity of quantum tomography from genuine non-Gaussian entanglement", Nature Communications 17 1, 373 (2025).

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[6] J Knörzer, X Liu, B F Schiffer, and J Tura, "Distributed quantum information processing: a review of recent progress", Reports on Progress in Physics 89 7, 074401 (2026).

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[8] Austen Couvertier and Ting Yu, "Entanglement dynamics of separable squeezed states in finite-memory structured reservoir", Physical Review A 113 6, 063724 (2026).

[9] Salman Beigi, Milad M. Goodarzi, and Hami Mehrabi, "Optimal Convergence Rates in Trace Distance and Relative Entropy for the Quantum Central Limit Theorem", Communications in Mathematical Physics 407 8, 176 (2026).

[10] Štěpán Šmíd, Richard Meister, Mario Berta, and Roberto Bondesan, "Rapid Mixing of Quantum Gibbs Samplers for Weakly-Interacting Quantum Systems", arXiv:2510.04954, (2025).

[11] Frank Ernesto Quintela Rodriguez, Francesco Anna Mele, Salvatore Francesco Emanuele Oliviero, Vittorio Giovannetti, Ludovico Lami, and Vasco Cavina, "Extracting energy via bosonic Gaussian operations", arXiv:2503.21748, (2025).

[12] Zixin Huang and Mark M. Wilde, "Information geometry of bosonic Gaussian thermal states", arXiv:2411.18268, (2024).

[13] Lennart Bittel, Antonio Anna Mele, Jens Eisert, and Lorenzo Leone, "Optimal Trace-Distance Bounds for Free-Fermionic States: Testing and Improved Tomography", PRX Quantum 6 3, 030341 (2025).

[14] Francesco Albarelli, Dominic Branford, and Jesús Rubio, "Measurement incompatibility in Bayesian multiparameter quantum estimation", arXiv:2511.16645, (2025).

[15] Marco Fanizza, Cambyse Rouzé, and Daniel Stilck França, "Efficient Hamiltonian, structure and trace distance learning of Gaussian states", arXiv:2411.03163, (2024).

[16] A. S. Holevo, "On estimates of trace-norm distance between quantum Gaussian states", arXiv:2408.11400, (2024).

[17] Vishnu Iyer, "Mildly-Interacting Fermionic Unitaries are Efficiently Learnable", arXiv:2504.11318, (2025).

[18] Francesco Anna Mele, Giovanni Barbarino, Vittorio Giovannetti, and Marco Fanizza, "Achievable rates in non-asymptotic bosonic quantum communication", arXiv:2502.05524, (2025).

[19] A. S. Holevo, "On estimates of the Bures distance between bosonic Gaussian states", arXiv:2412.04875, (2024).

[20] Riko Schadow, Naomi Spier, Stefan N. van den Hoven, Malaquias Correa Anguita, Redlef B. G. Braamhaar, Sara Marzban, Jens Eisert, Jelmer J. Renema, and Nathan Walk, "Certification of linear optical quantum state preparation", arXiv:2602.12269, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-09-08 14:05:10) and SAO/NASA ADS (last updated successfully 2026-09-08 14:05:13). The list may be incomplete as not all publishers provide suitable and complete citation data.

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