Metrology-assisted entanglement distribution in noisy quantum networks

Simon Morelli1,2, David Sauerwein3, Michalis Skotiniotis4, and Nicolai Friis1,2

1Institute for Quantum Optics and Quantum Information — IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
2Atominstitut, Technische Universität Wien, 1020 Vienna, Austria
3Amazon Web Services Europe, Zürich, Switzerland
4Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

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Abstract

We consider the distribution of high-dimensional entangled states to multiple parties via noisy channels and the subsequent probabilistic conversion of these states to desired target states using stochastic local operations and classical communication. We show that such state-conversion protocols can be enhanced by embedded channel-estimation routines at no additional cost in terms of the number of copies of the distributed states. The defining characteristic of our strategy is the use of those copies for which the conversion was unsuccessful for the estimation of the noise, thus allowing one to counteract its detrimental effect on the successfully converted copies. Although this idea generalizes to various more complex situations, we focus on the realistic scenario, where only finitely many copies are distributed and where the parties are not required to process multiple copies simultaneously. In particular, we investigate the performance of so-called one-successful-branch protocols, applied sequentially to single copies and an adaptive Bayesian estimation strategy. Finally, we compare our strategy to more general but less easily practically implementable strategies involving distillation and the use of quantum memories to process multiple copies simultaneously.

Entanglement shared between multiple users, for instance within a quantum network, is a crucial resource that allows one to overcome the restrictions of local operations and classical communication and thereby to implement classically impossible tasks. In this work we propose a novel strategy for the distribution of high-dimensional multipartite entanglement within noisy quantum networks with subsequent probabilistic state conversion. For certain types of noise the conversion can be carried out without the exact knowledge of the noise, which opens the possibility to use unsuccessfully converted copies for the estimation of the channel. Without sacrificing potentially good copies for this task, this protocol thus gains an advantage over comparable strategies in terms of the obtained number of copies close to the desired target state. We show the potential of the proposed strategy in a concrete example, where we consider the distribution of generic GHZ-type states in a network in the presence of local dephasing noise.

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► References

[1] Andrew J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions, Phys. Rev. A 69, 052330 (2004), arXiv:quant-ph/​0310137.
https:/​/​doi.org/​10.1103/​PhysRevA.69.052330
arXiv:quant-ph/0310137

[2] Tohya Hiroshima and Masahito Hayashi, Entanglement and Quantum Error Correction, in Quantum Computation and Information, edited by Hiroshi Imai and Masahito Hayashi (Springer, Berlin, Heidelberg, 2006) Chap. 5, pp. 111–132.
https:/​/​doi.org/​10.1007/​3-540-33133-6_5

[3] Wolfgang Dür and Hans J. Briegel, Entanglement purification and quantum error correction, Rep. Prog. Phys. 70, 1381 (2007), arXiv:0705.4165.
https:/​/​doi.org/​10.1088/​0034-4885/​70/​8/​R03
arXiv:0705.4165

[4] Todd A. Brun and Min-Hsiu Hsieh, Entanglement-assisted quantum error-correcting codes, in Quantum Error Correction, edited by Daniel A. Lidar and Todd A. Brun (Cambridge University Press, Cambridge, U.K., 2013) Chap. 7, pp. 181–200, arXiv:1610.04013.
https:/​/​doi.org/​10.1017/​CBO9781139034807.009
arXiv:1610.04013

[5] Oliver Marty, Marcus Cramer, and Martin B. Plenio, Practical Entanglement Estimation for Spin-System Quantum Simulators, Phys. Rev. Lett. 116, 105301 (2016), arXiv:1504.03572.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.105301
arXiv:1504.03572

[6] Nicolai Friis, Oliver Marty, Christine Maier, Cornelius Hempel, Milan Holzäpfel, Petar Jurcevic, Martin B. Plenio, Marcus Huber, Christian Roos, Rainer Blatt, and Ben Lanyon, Observation of Entangled States of a Fully Controlled 20-Qubit System, Phys. Rev. X 8, 021012 (2018), arXiv:1711.11092.
https:/​/​doi.org/​10.1103/​PhysRevX.8.021012
arXiv:1711.11092

[7] Marcello Dalmonte, Benoı̂t Vermersch, and Peter Zoller, Quantum simulation and spectroscopy of entanglement Hamiltonians, Nat. Phys. 14, 827 (2018), arXiv:1707.04455.
https:/​/​doi.org/​10.1038/​s41567-018-0151-7
arXiv:1707.04455

[8] Michael Epping, Hermann Kampermann, Chiara Macchiavello, and Dagmar Bruß, Multi-partite entanglement can speed up quantum key distribution in networks, New J. Phys. 19, 093012 (2017), arXiv:1612.05585.
https:/​/​doi.org/​10.1088/​1367-2630/​aa8487
arXiv:1612.05585

[9] Stefan Bäuml and Koji Azuma, Fundamental limitation on quantum broadcast networks, Quantum Sci. Technol. 2, 024004 (2017), arXiv:1609.03994.
https:/​/​doi.org/​10.1088/​2058-9565/​aa6d3c
arXiv:1609.03994

[10] Matej Pivoluska, Marcus Huber, and Mehul Malik, Layered quantum key distribution, Phys. Rev. A 97, 032312 (2018), arXiv:1709.00377.
https:/​/​doi.org/​10.1103/​PhysRevA.97.032312
arXiv:1709.00377

[11] Jérémy Ribeiro, Gláucia Murta, and Stephanie Wehner, Fully device-independent conference key agreement, Phys. Rev. A 97, 022307 (2018), arXiv:1708.00798.
https:/​/​doi.org/​10.1103/​PhysRevA.97.022307
arXiv:1708.00798

[12] Matteo Pompili, Sophie L. N. Hermans, Simon Baier, Hans K. C. Beukers, Peter C. Humphreys, Raymond N. Schouten, Raymond F. L. Vermeulen, Marijn J. Tiggelman, Laura dos Santos Martins, Bas Dirkse, Stephanie Wehner, and Ronald Hanson, Realization of a multinode quantum network of remote solid-state qubits, Science 372, 259 (2021), arXiv:2102.04471.
https:/​/​doi.org/​10.1126/​science.abg1919
arXiv:2102.04471

[13] H. Jeff Kimble, The quantum internet, Nature 453, 1023 (2008), arXiv:0806.4195.
https:/​/​doi.org/​10.1038/​nature07127
arXiv:0806.4195

[14] Stephanie Wehner, David Elkouss, and Ronald Hanson, Quantum internet: A vision for the road ahead, Science 362, eaam9288 (2018).
https:/​/​doi.org/​10.1126/​science.aam9288

[15] Angela Sara Cacciapuoti, Marcello Caleffi, Francesco Tafuri, Francesco Saverio Cataliotti, Stefano Gherardini, and Giuseppe Bianchi, Quantum Internet: Networking Challenges in Distributed Quantum Computing, IEEE Network 34, 137 (2020), arXiv:1810.08421.
https:/​/​doi.org/​10.1109/​MNET.001.1900092
arXiv:1810.08421

[16] Wolfgang Dür, Raphael Lamprecht, and Stefan Heusler, Towards a quantum internet, Eur. J. Phys. 38, 043001 (2017).
https:/​/​doi.org/​10.1088/​1361-6404/​aa6df7

[17] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels, Phys. Rev. Lett. 70, 1895–1899 (1993).
https:/​/​doi.org/​10.1103/​PhysRevLett.70.1895

[18] Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter, and Anton Zeilinger, Experimental Quantum Teleportation, Nature 390, 575–579 (1997), arXiv:1901.11004.
https:/​/​doi.org/​10.1038/​37539
arXiv:1901.11004

[19] Daniel Llewellyn, Yunhong Ding, Imad I. Faruque, Stefano Paesani, Davide Bacco, Raffaele Santagati, Yan-Jun Qian, Yan Li, Yun-Feng Xiao, Marcus Huber, Mehul Malik, Gary F. Sinclair, Xiaoqi Zhou, Karsten Rottwitt, Jeremy L. O'Brien, John G. Rarity, Qihuang Gong, Leif K. Oxenlowe, Jianwei Wang, and Mark G. Thompson, Chip-to-chip quantum teleportation and multi-photon entanglement in silicon, Nat. Phys. 16, 148 (2020), arXiv:1911.07839.
https:/​/​doi.org/​10.1038/​s41567-019-0727-x
arXiv:1911.07839

[20] Wolfgang Dür, Guifré Vidal, and Juan Ignacio Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000), arXiv:quant-ph/​0005115.
https:/​/​doi.org/​10.1103/​PhysRevA.62.062314
arXiv:quant-ph/0005115

[21] Antonio Acín, Dagmar Bruß, Maciej Lewenstein, and Anna Sanpera, Classification of Mixed Three-Qubit States, Phys. Rev. Lett. 87, 040401 (2001), arXiv:quant-ph/​0103025.
https:/​/​doi.org/​10.1103/​PhysRevLett.87.040401
arXiv:quant-ph/0103025

[22] Julio I. de Vicente, Cornelia Spee, and Barbara Kraus, Maximally Entangled Set of Multipartite Quantum States, Phys. Rev. Lett. 111, 110502 (2013), arXiv:1305.7398.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.110502
arXiv:1305.7398

[23] Katharina Schwaiger, David Sauerwein, Martí Cuquet, Julio I. de Vicente, and Barbara Kraus, Operational Multipartite Entanglement Measures, Phys. Rev. Lett. 115, 150502 (2015), arXiv:1503.00615.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.150502
arXiv:1503.00615

[24] Julio I. de Vicente, Cornelia Spee, David Sauerwein, and Barbara Kraus, Entanglement manipulation of multipartite pure states with finite rounds of classical communication, Phys. Rev. A 95, 012323 (2017), arXiv:1607.05145.
https:/​/​doi.org/​10.1103/​PhysRevA.95.012323
arXiv:1607.05145

[25] Cornelia Spee, Julio I. de Vicente, David Sauerwein, and Barbara Kraus, Entangled Pure State Transformations via Local Operations Assisted by Finitely Many Rounds of Classical Communication, Phys. Rev. Lett. 118, 040503 (2017), arXiv:1606.04418.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.040503
arXiv:1606.04418

[26] David Sauerwein, Nolan R. Wallach, Gilad Gour, and Barbara Kraus, Transformations among Pure Multipartite Entangled States via Local Operations are Almost Never Possible, Phys. Rev. X 8, 031020 (2018a), arXiv:1711.11056.
https:/​/​doi.org/​10.1103/​PhysRevX.8.031020
arXiv:1711.11056

[27] Eric Chitambar, Debbie Leung, Laura Mančinska, Maris Ozols, and Andreas Winter, Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask), Commun. Math. Phys. 328, 303 (2014), arXiv:1210.4583.
https:/​/​doi.org/​10.1007/​s00220-014-1953-9
arXiv:1210.4583

[28] Jessica Bavaresco, Natalia Herrera Valencia, Claude Klöckl, Matej Pivoluska, Paul Erker, Nicolai Friis, Mehul Malik, and Marcus Huber, Measurements in two bases are sufficient for certifying high-dimensional entanglement, Nat. Phys. 14, 1032 (2018), arXiv:1709.07344.
https:/​/​doi.org/​10.1038/​s41567-018-0203-z
arXiv:1709.07344

[29] He Lu, Qi Zhao, Zheng-Da Li, Xu-Fei Yin, Xiao Yuan, Jui-Chen Hung, Luo-Kan Chen, Li Li, Nai-Le Liu, Cheng-Zhi Peng, Yeong-Cherng Liang, Xiongfeng Ma, Yu-Ao Chen, and Jian-Wei Pan, Entanglement Structure: Entanglement Partitioning in Multipartite Systems and Its Experimental Detection Using Optimizable Witnesses, Phys. Rev. X 8, 021072 (2018), arXiv:1711.01784.
https:/​/​doi.org/​10.1103/​PhysRevX.8.021072
arXiv:1711.01784

[30] Nicolai Friis, Giuseppe Vitagliano, Mehul Malik, and Marcus Huber, Entanglement Certification From Theory to Experiment, Nat. Rev. Phys. 1, 72 (2019), arXiv:1906.10929.
https:/​/​doi.org/​10.1038/​s42254-018-0003-5
arXiv:1906.10929

[31] You Zhou, Qi Zhao, Xiao Yuan, and Xiongfeng Ma, Detecting multipartite entanglement structure with minimal resources, npj Quantum Inf. 5, 83 (2019), arXiv:1904.05001.
https:/​/​doi.org/​10.1038/​s41534-019-0200-9
arXiv:1904.05001

[32] Natalia Herrera Valencia, Vatshal Srivastav, Matej Pivoluska, Marcus Huber, Nicolai Friis, Will McCutcheon, and Mehul Malik, High-Dimensional Pixel Entanglement: Efficient Generation and Certification, Quantum 4, 376 (2020), arXiv:2004.04994.
https:/​/​doi.org/​10.22331/​q-2020-12-24-376
arXiv:2004.04994

[33] Gary J. Mooney, Gregory A. L. White, Charles D. Hill, and Lloyd C. L. Hollenberg, Whole-Device Entanglement in a 65-Qubit Superconducting Quantum Computer, Adv. Quantum Technol. 4, 2100061 (2021a), arXiv:2102.11521.
https:/​/​doi.org/​10.1002/​qute.202100061
arXiv:2102.11521

[34] Raju Valivarthi, Samantha Davis, Cristian Pena, Si Xie, Nikolai Lauk, Lautaro Narvaez, Jason P. Allmaras, Andrew D. Beyer, Yewon Gim, Meraj Hussein, George Iskander, Hyunseong Linus Kim, Boris Korzh, Andrew Mueller, Mandy Rominsky, Matthew Shaw, Dawn Tang, Emma E. Wollman, Christoph Simon, Panagiotis Spentzouris, Neil Sinclair, Daniel Oblak, and Maria Spiropulu, Teleportation Systems Towards a Quantum Internet, PRX Quantum 1, 020317 (2020), arXiv:2007.11157.
https:/​/​doi.org/​10.1103/​PRXQuantum.1.020317
arXiv:2007.11157

[35] Miguel Navascues, Elie Wolfe, Denis Rosset, and Alejandro Pozas-Kerstjens, Genuine Network Multipartite Entanglement, Phys. Rev. Lett. 125, 240505 (2020), arXiv:2002.02773.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.240505
arXiv:2002.02773

[36] Tristan Kraft, Sébastien Designolle, Christina Ritz, Nicolas Brunner, Otfried Gühne, and Marcus Huber, Quantum entanglement in the triangle network, Phys. Rev. A 103, L060401 (2021a), arXiv:2002.03970.
https:/​/​doi.org/​10.1103/​PhysRevA.103.L060401
arXiv:2002.03970

[37] Tristan Kraft, Cornelia Spee, Xiao-Dong Yu, and Otfried Gühne, Characterizing quantum networks: Insights from coherence theory, Phys. Rev. A 103, 052405 (2021b), arXiv:2006.06693.
https:/​/​doi.org/​10.1103/​PhysRevA.103.052405
arXiv:2006.06693

[38] Johan Åberg, Ranieri Nery, Cristhiano Duarte, and Rafael Chaves, Semidefinite tests for quantum network topologies, Phys. Rev. Lett. 125, 110505 (2020), arXiv:2002.05801.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.110505
arXiv:2002.05801

[39] Cornelia Spee and Tristan Kraft, Transformations in quantum networks via local operations assisted by finitely many rounds of classical communication, arXiv.2105.01090 [quant-ph] (2021).
https:/​/​doi.org/​10.48550/​arXiv.2105.01090

[40] Hayata Yamasaki, Simon Morelli, Markus Miethlinger, Jessica Bavaresco, Nicolai Friis, and Marcus Huber, Activation of genuine multipartite entanglement: beyond the single-copy paradigm of entanglement characterisation, Quantum 6, 695 (2022), arXiv:2106.01372.
https:/​/​doi.org/​10.22331/​q-2022-04-25-695
arXiv:2106.01372

[41] Antoine Neven, David Gunn, Martin Hebenstreit, and Barbara Kraus, Local Transformations of Multiple Multipartite States, SciPost Phys. 11, 42 (2021), arXiv:2007.06256.
https:/​/​doi.org/​10.21468/​SciPostPhys.11.2.042
arXiv:2007.06256

[42] Gilad Gour, Barbara Kraus, and Nolan R. Wallach, Almost all multipartite qubit quantum states have trivial stabilizer, J. Math. Phys. 58, 092204 (2017), arXiv:1609.01327.
https:/​/​doi.org/​10.1063/​1.5003015
arXiv:1609.01327

[43] Antonio Acín, Enric Jané, Wolfgang Dür, and Guifre Vidal, Optimal Distillation of a Greenberger-Horne-Zeilinger State, Phys. Rev. Lett. 85, 4811 (2000), arXiv:quant-ph/​0007042.
https:/​/​doi.org/​10.1103/​PhysRevLett.85.4811
arXiv:quant-ph/0007042

[44] David Sauerwein, Katharina Schwaiger, and Barbara Kraus, Discrete and differentiable entanglement transformations, arXiv:1808.02819 [quant-ph] (2018b).
https:/​/​doi.org/​10.48550/​arXiv.1808.02819
arXiv:1808.02819

[45] Marcin Jarzyna, Konrad Banaszek, and Rafał Demkowicz-Dobrzański, Dephasing in coherent communication with weak signal states, J. Phys. A: Math. Theor. 47, 275302 (2014), arXiv:1307.6871.
https:/​/​doi.org/​10.1088/​1751-8113/​47/​27/​275302
arXiv:1307.6871

[46] Keith H. Wanser, Fundamental phase noise limit in optical fibres due to temperature fluctuations, Electron. Lett. 28, 53 (1992).
https:/​/​doi.org/​10.1049/​el:19920033

[47] Marco Fanizza, Matteo Rosati, Michalis Skotiniotis, John Calsamiglia, and Vittorio Giovannetti, Squeezing-enhanced communication without a phase reference, Quantum 5, 608 (2021), arXiv:2006.06522.
https:/​/​doi.org/​10.22331/​q-2021-12-23-608
arXiv:2006.06522

[48] Guifré Vidal, Entanglement of Pure States for a Single Copy, Phys. Rev. Lett. 83, 1046 (1999), arXiv:quant-ph/​9902033.
https:/​/​doi.org/​10.1103/​PhysRevLett.83.1046
arXiv:quant-ph/9902033

[49] Xi-Lin Wang, Yi-Han Luo, He-Liang Huang, Ming-Cheng Chen, Zu-En Su, Chang Liu, Chao Chen, Wei Li, Yu-Qiang Fang, Xiao Jiang, Jun Zhang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan, 18-Qubit Entanglement with Six Photons' Three Degrees of Freedom, Phys. Rev. Lett. 120, 260502 (2018), arXiv:1801.04043.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.260502
arXiv:1801.04043

[50] Ming Gong, Ming-Cheng Chen, Yarui Zheng, Shiyu Wang, Chen Zha, Hui Deng, Zhiguang Yan, Hao Rong, Yulin Wu, Shaowei Li, Fusheng Chen, Youwei Zhao, Futian Liang, Jin Lin, Yu Xu, Cheng Guo, Lihua Sun, Anthony D. Castellano, Haohua Wang, Chengzhi Peng, Chao-Yang Lu, Xiaobo Zhu, and Jian-Wei Pan, Genuine 12-Qubit Entanglement on a Superconducting Quantum Processor, Phys. Rev. Lett. 122, 110501 (2019), arXiv:1811.02292.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.110501
arXiv:1811.02292

[51] Gary J. Mooney, Gregory A. L. White, Charles D. Hill, and Lloyd C. L. Hollenberg, Generation and verification of 27-qubit Greenberger-Horne-Zeilinger states in a superconducting quantum computer, J. Phys. Commun. 5, 095004 (2021b), arXiv:2101.08946.
https:/​/​doi.org/​10.1088/​2399-6528/​ac1df7
arXiv:2101.08946

[52] Ivan Pogorelov, Thomas Feldker, Christian D. Marciniak, Georg Jacob, Verena Podlesnic, Michael Meth, Vlad Negnevitsky, Martin Stadler, Kirill Lakhmanskiy, Rainer Blatt, Philipp Schindler, and Thomas Monz, Compact Ion-Trap Quantum Computing Demonstrator, PRX Quantum 2, 020343 (2021), arXiv:2101.11390.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.020343
arXiv:2101.11390

[53] Nilanjana Datta and Felix Leditzky, Second-Order Asymptotics for Source Coding, Dense Coding, and Pure-State Entanglement Conversions, IEEE Trans. Inf. Theory 61, 582 (2015), arXiv:1403.2543.
https:/​/​doi.org/​10.1109/​TIT.2014.2366994
arXiv:1403.2543

[54] Kun Fang, Xin Wang, Marco Tomamichel, and Runyao Duan, Non-Asymptotic Entanglement Distillation, IEEE Trans. Inf. Theory 65, 6454 (2019), arXiv:1706.06221.
https:/​/​doi.org/​10.1109/​TIT.2019.2914688
arXiv:1706.06221

[55] Stephen Brierley, Stefan Weigert, and Ingemar Bengtsson, All Mutually Unbiased Bases in Dimensions Two to Five, Quantum Inf. Comput. 10, 0803 (2009), arXiv:0907.4097.
https:/​/​doi.org/​10.26421/​QIC10.9-10-6
arXiv:0907.4097

[56] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, and Benjamin Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53, 2046 (1996), arXiv:quant-ph/​9511030.
https:/​/​doi.org/​10.1103/​physreva.53.2046
arXiv:quant-ph/9511030

[57] John A. Smolin, Frank Verstraete, and Andreas Winter, Entanglement of assistance and multipartite state distillation, Phys. Rev. A 72, 052317 (2005), arXiv:quant-ph/​0505038.
https:/​/​doi.org/​10.1103/​PhysRevA.72.052317
arXiv:quant-ph/0505038

[58] Alexander Streltsov, Clément Meignant, and Jens Eisert, Rates of Multipartite Entanglement Transformations, Phys. Rev. Lett. 125, 080502 (2020), arXiv:1709.09693.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.080502
arXiv:1709.09693

[59] Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti, and Massimiliano Federico Sacchi, Efficient Use of Quantum Resources for the Transmission of a Reference Frame, Phys. Rev. Lett. 93, 180503 (2004), arXiv:quant-ph/​0405095.
https:/​/​doi.org/​10.1103/​PhysRevLett.93.180503
arXiv:quant-ph/0405095

[60] Jonas Kahn, Fast rate estimation of a unitary operation in $\text{SU}(d)$, Phys. Rev. A 75, 022326 (2007), arXiv:quant-ph/​0603115.
https:/​/​doi.org/​10.1103/​PhysRevA.75.022326
arXiv:quant-ph/0603115

[61] Richard D. Gill and Boris Y. Levit, Applications of the van Trees Inequality: A Bayesian Cramér-Rao Bound, Bernoulli 1, 59 (1995), https:/​/​projecteuclid.org/​euclid.bj/​1186078362.
https:/​/​doi.org/​10.2307/​3318681

[62] Samuel L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
https:/​/​doi.org/​10.1103/​PhysRevLett.72.3439

[63] Rafał Demkowicz-Dobrzański, Marcin Jarzyna, and Janek Kołodyński, Quantum limits in optical interferometry, Prog. Optics 60, 345 (2015), arXiv:1405.7703.
https:/​/​doi.org/​10.1016/​bs.po.2015.02.003
arXiv:1405.7703

[64] Nicolai Friis, Davide Orsucci, Michalis Skotiniotis, Pavel Sekatski, Vedran Dunjko, Hans J. Briegel, and Wolfgang Dür, Flexible resources for quantum metrology, New J. Phys. 19, 063044 (2017), arXiv:1610.09999.
https:/​/​doi.org/​10.1088/​1367-2630/​aa7144
arXiv:1610.09999

[65] Simon Morelli, Ayaka Usui, Eliza Agudelo, and Nicolai Friis, Bayesian parameter estimation using Gaussian states and measurements, Quantum Sci. Technol. 6, 025018 (2021), arXiv:2009.03709.
https:/​/​doi.org/​10.1088/​2058-9565/​abd83d
arXiv:2009.03709

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