Imaginarity-free quantum multiparameter estimation

Jisho Miyazaki; Keiji Matsumoto

doi:10.22331/q-2022-03-10-665

Imaginarity-free quantum multiparameter estimation

Jisho Miyazaki¹ and Keiji Matsumoto²

¹Shinpouin, Mikataharacho, Kita-ku, Hamamatsu, Shizuoka, 433-8105, Japan
²Quantum Computation Group, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430

Published:	2022-03-10, volume 6, page 665
Eprint:	arXiv:2010.15465v3
Doi:	https://doi.org/10.22331/q-2022-03-10-665
Citation:	Quantum 6, 665 (2022).

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Abstract

Multiparameter quantum estimation is made difficult by the following three obstacles. First, incompatibility among different physical quantities poses a limit on the attainable precision. Second, the ultimate precision is not saturated until you discover the optimal measurement. Third, the optimal measurement may generally depend on the target values of parameters, and thus may be impossible to perform for unknown target states.
We present a method to circumvent these three obstacles. A class of quantum statistical models, which utilizes antiunitary symmetries or, equivalently, real density matrices, offers compatible multiparameter estimations. The symmetries accompany the target-independent optimal measurements for pure-state models. Based on this finding, we propose methods to implement antiunitary symmetries for quantum metrology schemes. We further introduce a function which measures antiunitary asymmetry of quantum statistical models as a potential tool to characterize quantumness of phase transitions.

► BibTeX data

@article{Miyazaki2022imaginarityfree,
  doi = {10.22331/q-2022-03-10-665},
  url = {https://doi.org/10.22331/q-2022-03-10-665},
  title = {Imaginarity-free quantum multiparameter estimation},
  author = {Miyazaki, Jisho and Matsumoto, Keiji},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {665},
  month = mar,
  year = {2022}
}

► References

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

[2] V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photon. 5, 222 (2011).
https://doi.org/10.1038/nphoton.2011.35

[3] R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, Prog. Opt. 60, 345 (2015).
https://doi.org/10.1016/bs.po.2015.02.003

[4] C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017).
https://doi.org/10.1103/RevModPhys.89.035002

[5] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Rev. Mod. Phys. 90, 035005 (2018).
https://doi.org/10.1103/RevModPhys.90.035005

[6] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, Rev. Mod. Phys. 90, 035006 (2018).
https://doi.org/10.1103/RevModPhys.90.035006

[7] S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Nat. Photonics 12, 724 (2018).
https://doi.org/10.1038/s41566-018-0301-6

[8] I. R. Berchera and I. P. Degiovanni, Metrologia 56, 024001 (2019).
https://doi.org/10.1088/1681-7575/aaf7b2

[9] C. Helstrom, Phys. Lett. A 25, 101 (1967).
https://doi.org/10.1016/0375-9601(67)90366-0

[10] C. Helstrom, IEEE Trans. Inf. Theory 14, 234 (1968).
https://doi.org/10.1109/TIT.1968.1054108

[11] A. S. Holevo, Probabilistic and statistical aspects of quantum theory, Vol. 1 (Springer Science & Business Media, 2011).
https://doi.org/10.1007/978-88-7642-378-9

[12] M. Hayashi, Quantum Information Theory, Graduate Texts in Physics, Springer (Springer, 2017).
https://doi.org/10.1007/978-3-662-49725-8

[13] M. Szczykulska, T. Baumgratz, and A. Datta, Adv. Phys. X 1, 621 (2016).
https://doi.org/10.1080/23746149.2016.1230476

[14] F. Albarelli, J. F. Friel, and A. Datta, Phys. Rev. Lett. 123, 200503 (2019).
https://doi.org/10.1103/PhysRevLett.123.200503

[15] T. Baumgratz and A. Datta, Phys. Rev. Lett. 116, 030801 (2016).
https://doi.org/10.1103/PhysRevLett.116.030801

[16] P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, Phys. Rev. Lett. 111, 070403 (2013).
https://doi.org/10.1103/PhysRevLett.111.070403

[17] C. N. Gagatsos, D. Branford, and A. Datta, Phys. Rev. A 94, 042342 (2016).
https://doi.org/10.1103/PhysRevA.94.042342

[18] M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, M. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, Nat. Commun. 5, 1 (2014).
https://doi.org/10.1038/ncomms4532

[19] E. Roccia, V. Cimini, M. Sbroscia, I. Gianani, L. Ruggiero, L. Mancino, M. G. Genoni, M. A. Ricci, and M. Barbieri, Optica 5, 1171 (2018).
https://doi.org/10.1364/OPTICA.5.001171

[20] M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
https://doi.org/10.1103/PhysRevLett.121.250503

[21] E. Polino, M. Riva, M. Valeri, R. Silvestri, G. Corrielli, A. Crespi, N. Spagnolo, R. Osellame, and F. Sciarrino, Optica 6, 288 (2019).
https://doi.org/10.1364/OPTICA.6.000288

[22] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, 1976).

[23] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North Holland, 1982).
https://doi.org/10.1007/978-88-7642-378-9

[24] P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, Phys. Rev. A 89, 023845 (2014).
https://doi.org/10.1103/PhysRevA.89.023845

[25] K. Matsumoto, J. Phys. A: Math. Gen. 35, 3111 (2002).
https://doi.org/10.1088/0305-4470/35/13/307

[26] C. Vaneph, T. Tufarelli, and M. G. Genoni, Quantum Meas. Quantum Metrol. 1, 12 (25 Jun. 2013).
https://doi.org/10.2478/qmetro-2013-0003

[27] J. Suzuki, J. Math. Phys. 57, 042201 (2016).
https://doi.org/10.1063/1.4945086

[28] S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Phys. Rev. A 94, 052108 (2016).
https://doi.org/10.1103/PhysRevA.94.052108

[29] L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, Phys. Rev. Lett. 119, 130504 (2017).
https://doi.org/10.1103/PhysRevLett.119.130504

[30] J. Yang, S. Pang, Y. Zhou, and A. N. Jordan, Phys. Rev. A 100, 032104 (2019a).
https://doi.org/10.1103/PhysRevA.100.032104

[31] C. Napoli, S. Piano, R. Leach, G. Adesso, and T. Tufarelli, Phys. Rev. Lett. 122, 140505 (2019).
https://doi.org/10.1103/PhysRevLett.122.140505

[32] S. Kukita, J. Phys. A: Math. Theor. 53, 095303 (2020).
https://doi.org/10.1088/1751-8121/ab6d3d

[33] F. Belliardo and V. Giovannetti, New J. Phys. 23, 063055 (2021).
https://doi.org/10.1088/1367-2630/ac04ca

[34] A. S. Holevo, in Proceedings of the Third Japan — USSR Symposium on Probability Theory, edited by G. Maruyama and J. V. Prokhorov (Springer Berlin Heidelberg, Berlin, Heidelberg, 1976) pp. 194–222.
https://doi.org/10.1007/BFb0077491

[35] M. Guţă and J. Kahn, Phys. Rev. A 73, 052108 (2006).
https://doi.org/10.1103/PhysRevA.73.052108

[36] M. Hayashi and K. Matsumoto, J. Math. Phys. 49, 102101 (2008).
https://doi.org/10.1063/1.2988130

[37] J. Kahn and M. Guţă, Comm. Math. Phys 289, 597 (2009).
https://doi.org/10.1007/s00220-009-0787-3

[38] K. Yamagata, A. Fujiwara, and R. D. Gill, Ann. Stat. 41, 2197 (2013).
https://doi.org/10.1214/13-AOS1147

[39] Y. Yang, G. Chiribella, and M. Hayashi, Comm. Math. Phys. 368, 223 (2019b).
https://doi.org/10.1007/s00220-019-03433-4

[40] M. Bradshaw, S. M. Assad, and P. K. Lam, Phys. Lett. A 381, 2598 (2017).
https://doi.org/10.1016/j.physleta.2017.06.024

[41] M. Bradshaw, P. K. Lam, and S. M. Assad, Phys. Rev. A 97, 012106 (2018).
https://doi.org/10.1103/PhysRevA.97.012106

[42] W. Górecki, S. Zhou, L. Jiang, and R. Demkowicz-Dobrzański, Quantum 4, 288 (2020).
https://doi.org/10.22331/q-2020-07-02-288

[43] A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, J. Stat. Mech. 2019, 094010 (2019).
https://doi.org/10.1088/1742-5468/ab3ccb

[44] K. Yamagata, J. Math. Phys. 62, 062203 (2021).
https://doi.org/10.1063/5.0047496

[45] J. S. Sidhu, Y. Ouyang, E. T. Campbell, and P. Kok, Phys. Rev. X 11, 011028 (2021).
https://doi.org/10.1103/PhysRevX.11.011028

[46] A. Z. Goldberg, L. L. Sánchez-Soto, and H. Ferretti, Phys. Rev. Lett. 127, 110501 (2021).
https://doi.org/10.1103/PhysRevLett.127.110501

[47] M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).
https://doi.org/10.1103/PhysRevX.6.031033

[48] J. Řeháček, Z. Hradil, D. Koutný, J. Grover, A. Krzic, and L. L. Sánchez-Soto, Phys. Rev. A 98, 012103 (2018).
https://doi.org/10.1103/PhysRevA.98.012103

[49] Y. L. Len, arXiv preprint arXiv:2109.07430 (2021).
https://doi.org/10.48550/arXiv.2109.07430
arXiv:2109.07430

[50] J. de Almeida, M. Lewenstein, and M. Skotiniotis, arXiv preprint arXiv:2110.00986 (2021).
https://doi.org/10.48550/arXiv.2110.00986
arXiv:2110.00986

[51] H. Lee, P. Kok, and J. P. Dowling, J. Mod. Opt. 49, 2325 (2002).
https://doi.org/10.1080/0950034021000011536

[52] V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004).
https://doi.org/10.1126/science.1104149

[53] N. Gisin and S. Popescu, Phys. Rev. Lett. 83, 432 (1999).
https://doi.org/10.1103/PhysRevLett.83.432

[54] L. Chang, N. Li, S. Luo, and H. Song, Phys. Rev. A 89, 042110 (2014).
https://doi.org/10.1103/PhysRevA.89.042110

[55] A. Carollo, B. Spagnolo, and D. Valenti, Sci. Rep. 8, 1 (2018a).
https://doi.org/10.1038/s41598-018-27362-9

[56] A. Carollo, B. Spagnolo, and D. Valenti, Entropy 20, 485 (2018b).
https://doi.org/10.3390/e20070485

[57] L. Leonforte, D. Valenti, B. Spagnolo, A. A. Dubkov, and A. Carollo, J. Stat. Mech. 2019, 094001 (2019).
https://doi.org/10.1088/1742-5468/ab33f8

[58] F. Bascone, L. Leonforte, D. Valenti, B. Spagnolo, and A. Carollo, J. Stat. Mech. 2019, 094002 (2019a).
https://doi.org/10.1088/1742-5468/ab35e9

[59] F. Bascone, L. Leonforte, D. Valenti, B. Spagnolo, and A. Carollo, Phys. Rev. B 99, 205155 (2019b).
https://doi.org/10.1103/PhysRevB.99.205155

[60] A. Hickey and G. Gour, J. Phys. A: Math. Theor. 51, 414009 (2018).
https://doi.org/10.1088/1751-8121/aabe9c

[61] K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Phys. Rev. Lett. 126, 090401 (2021a).
https://doi.org/10.1103/PhysRevLett.126.090401

[62] K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Phys. Rev. A 103, 032401 (2021b).
https://doi.org/10.1103/PhysRevA.103.032401

[63] S. Amari and H. Nagaoka, Methods of Information Geometry, Translations of Mathematical Monographs, Vol. 191 (American Mathematical Society, 2000).
https://doi.org/10.1090/mmono/191

[64] J. Yang, private communication (2021).

[65] A. Uhlmann, Sci. China Phys. Mech. Astron. 59, 630301 (2016).
https://doi.org/10.1007/s11433-015-5777-1

[66] A. Fujiwara, Phys. Rev. A 65, 012316 (2001).
https://doi.org/10.1103/PhysRevA.65.012316

[67] J. Casanova, C. Sabín, J. León, I. L. Egusquiza, R. Gerritsma, C. F. Roos, J. J. García-Ripoll, and E. Solano, Phys. Rev. X 1, 021018 (2011).
https://doi.org/10.1103/PhysRevX.1.021018

[68] R. Di Candia, B. Mejia, H. Castillo, J. S. Pedernales, J. Casanova, and E. Solano, Phys. Rev. Lett. 111, 240502 (2013).
https://doi.org/10.1103/PhysRevLett.111.240502

[69] X. Zhang, Y. Shen, J. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, Nat. Commun. 6, 1 (2015).
https://doi.org/10.1038/ncomms8917

[70] M.-C. Chen, D. Wu, Z.-E. Su, X.-D. Cai, X.-L. Wang, T. Yang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, Phys. Rev. Lett. 116, 070502 (2016).
https://doi.org/10.1103/PhysRevLett.116.070502

[71] J. C. Loredo, M. P. Almeida, R. Di Candia, J. S. Pedernales, J. Casanova, E. Solano, and A. G. White, Phys. Rev. Lett. 116, 070503 (2016).
https://doi.org/10.1103/PhysRevLett.116.070503

[72] X.-H. Cheng, I. n. Arrazola, J. S. Pedernales, L. Lamata, X. Chen, and E. Solano, Phys. Rev. A 95, 022305 (2017).
https://doi.org/10.1103/PhysRevA.95.022305

[73] K. Matsumoto, A Geometrical Approach to Quantum Estimation Theory, Ph.D. thesis, The University of Tokyo (1997), preprint avairable at https://doi.org/10.48550/arXiv.2111.09667.
https://doi.org/10.11501/3163169

[74] K. Matsumoto, ``A geometrical approach to quantum estimation theory,'' in Asymptotic Theory of Quantum Statistical Inference, edited by M. Hayashi (WORLD SCIENTIFIC, 2005) Chap. 20, pp. 305–350.
https://doi.org/10.1142/9789812563071_0021

[75] E. C. G. Stueckelberg, Helv. Phys. Acta 33, 458 (1960).

[76] P. J. Lahti and M. J. Maczynski, J. Math. Phys. 28, 1764 (1987).
https://doi.org/10.1063/1.527822

[77] X.-M. Lu and X. Wang, Phys. Rev. Lett. 126, 120503 (2021).
https://doi.org/10.1103/PhysRevLett.126.120503

[78] A. Uhlmann, Rep. Math. Phys. 24, 229 (1986).
https://doi.org/10.1016/0034-4877(86)90055-8

[79] S. Massar, Phys. Rev. A 62, 040101(R) (2000).
https://doi.org/10.1103/PhysRevA.62.040101

[80] E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, AVS Quantum Sci. 2, 024703 (2020).
https://doi.org/10.1116/5.0007577

[81] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
https://doi.org/10.1103/PhysRevLett.69.2881

[82] J. A. Gross and C. M. Caves, J. Phys. A: Math. Theor. 54, 014001 (2020).
https://doi.org/10.1088/1751-8121/abb9ed

[83] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
https://doi.org/10.1103/PhysRevLett.72.3439

[84] S. L. Braunstein, C. M. Caves, and G. Milburn, Ann. Phys. 247, 135 (1996).
https://doi.org/10.1006/aphy.1996.0040

[85] N. J. Cerf and S. Iblisdir, Phys. Rev. Lett. 87, 247903 (2001).
https://doi.org/10.1103/PhysRevLett.87.247903

[86] S. L. Braunstein, S. Ghosh, and S. Severini, J. Phys. A: Math. Theor. 40, 1809 (2007).
https://doi.org/10.1088/1751-8113/40/8/009

[87] G. Kato, Phys. Rev. A 79, 032315 (2009).
https://doi.org/10.1103/PhysRevA.79.032315

[88] J. Miyazaki, arXiv preprint arXiv:2005.06685 (2020).
https://doi.org/10.48550/arXiv.2005.06685
arXiv:2005.06685

[89] J.-F. Tang, Z. Hou, J. Shang, H. Zhu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 124, 060502 (2020).
https://doi.org/10.1103/PhysRevLett.124.060502

[90] E. Bagan, M. Baig, A. Brey, R. Muñoz Tapia, and R. Tarrach, Phys. Rev. Lett. 85, 5230 (2000).
https://doi.org/10.1103/PhysRevLett.85.5230

[91] R. Jozsa and J. Schlienz, Phys. Rev. A 62, 012301 (2000).
https://doi.org/10.1103/PhysRevA.62.012301

[92] T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).
https://doi.org/10.1103/PhysRevLett.113.140401

[93] E. Chitambar and G. Gour, Phys. Rev. Lett. 117, 030401 (2016).
https://doi.org/10.1103/PhysRevLett.117.030401

[94] A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016).
https://doi.org/10.1103/PhysRevLett.116.120404

[95] S. Designolle, R. Uola, K. Luoma, and N. Brunner, Phys. Rev. Lett. 126, 220404 (2021).
https://doi.org/10.1103/PhysRevLett.126.220404

[96] A. Chrostowski, R. Demkowicz-Dobrzański, M. Jarzyna, and K. Banaszek, Int. J. Quantum Inf. 15, 1740005 (2017).
https://doi.org/10.1142/S0219749917400056

[97] J. Řeháček, Z. Hradil, B. Stoklasa, M. Paúr, J. Grover, A. Krzic, and L. L. Sánchez-Soto, Phys. Rev. A 96, 062107 (2017).
https://doi.org/10.1103/PhysRevA.96.062107

[98] Z. Yu and S. Prasad, Phys. Rev. Lett. 121, 180504 (2018).
https://doi.org/10.1103/PhysRevLett.121.180504

[99] J. Suzuki, Entropy 21, 703 (2019).
https://doi.org/10.3390/e21070703

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[2] Francisco Delgado, "Symmetries of Quantum Fisher Information as Parameter Estimator for Pauli Channels under Indefinite Causal Order", Symmetry 14 9, 1813 (2022).

[3] Lijun Peng and Xiao-Ming Lu, "Generalization of Rayleigh's criterion on parameter estimation with incoherent sources", Physical Review A 103 4, 042601 (2021).

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