Holevo Cramér-Rao bound: How close can we get without entangling measurements?

Aritra Das; Lorcán O. Conlon; Jun Suzuki; Simon K. Yung; Ping K. Lam; Syed M. Assad

doi:10.22331/q-2025-09-30-1867

Holevo Cramér-Rao bound: How close can we get without entangling measurements?

Aritra Das¹, Lorcán O. Conlon², Jun Suzuki³, Simon K. Yung^1,2, Ping K. Lam^2,1, and Syed M. Assad^2,1

¹Centre for Quantum Computation and Communication Technology, Department of Quantum Science and Technology, Australian National University, Canberra, ACT 2601, Australia
²Quantum Innovation Centre (Q.InC), Agency for Science Technology and Research (A*STAR), 2 Fusionopolis Way, Innovis 08-03, Singapore 138634, Singapore
³Graduate School of Informatics and Engineering, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan

Published:	2025-09-30, volume 9, page 1867
Editor:	Aaron Goldberg
Eprint:	arXiv:2405.09622v2
Doi:	https://doi.org/10.22331/q-2025-09-30-1867
Citation:	Quantum 9, 1867 (2025).

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Abstract

In multi-parameter quantum metrology, the resource of entanglement can lead to an increase in efficiency of the estimation process. Entanglement can be used in the state preparation stage, or the measurement stage, or both, to harness this advantage; here we focus on the role of entangling measurements. Specifically, entangling or collective measurements over multiple identical copies of a probe state are known to be superior to measuring each probe individually, but the extent of this improvement is an open problem. It is also known that such entangling measurements, though resource-intensive, are required to attain the ultimate limits in multi-parameter quantum metrology and quantum information processing tasks. In this work we investigate the maximum precision improvement that collective quantum measurements can offer over individual measurements for estimating parameters of qudit states, calling this the 'collective quantum enhancement'. We show that, whereas the maximum enhancement can, in principle, be a factor of $n$ for estimating $n$ parameters, this bound is not tight for large $n$. Instead, our results prove an enhancement linear in dimension of the qudit is possible using collective measurements and lead us to conjecture that this is the maximum collective quantum enhancement in any local estimation scenario.

Featured image: Metrological enhancement from entangling measurements versus separable measurements in qudit state estimation

Popular summary

The ultimate limits of precision can be attained by quantum theory by incorporating entanglement in the process. However, entangling measurements, which form a key part of this process, are extremely challenging to implement in practice, as reflected in the current state-of-the-art. We quantify the extent to which precision is lost by performing non-entangling measurements instead, showing this to be limited by the dimension of the quantum state the estimation is performed on.

The estimation of parameters of a quantum state is a routine component of several quantum protocols across computation, communication and cryptography. When such protocols are executed in real-world scenarios, the resources available are of course finite, but the most widely-used tool to quantify the precision of estimates assumes that entangling operations on infinitely-many copies of the probe state are possible. By identifying a situation where entangling measurements are the most superior to non-entangling measurements, we show that the precision prescribed thus can be arbitrarily far from the practically attainable precision, where only a few copies or even none are entangled before measurement.

Our work advocates a judicious choice of measurement strategy and corresponding performance metric when designing practical quantum estimation tasks or benchmarking the performance of real-world quantum measurements. Further investigating the advantage of entangling measurements in the finite-resource setting would endow us with a better understanding of the capabilities of quantum theory.

► BibTeX data

@article{Das2025holevocramerrao,
  doi = {10.22331/q-2025-09-30-1867},
  url = {https://doi.org/10.22331/q-2025-09-30-1867},
  title = {Holevo {C}ram{\'{e}}r-{R}ao bound: {H}ow close can we get without entangling measurements?},
  author = {Das, Aritra and Conlon, Lorc{\'{a}}n O. and Suzuki, Jun and Yung, Simon K. and Lam, Ping K. and Assad, Syed M.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1867},
  month = sep,
  year = {2025}
}

► 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. Photonics 5, 222–229 (2011).
https://doi.org/10.1038/nphoton.2011.35

[3] M. Szczykulska, T. Baumgratz, and A. Datta, Adv. Phys.: X 1, 621–639 (2016).
https://doi.org/10.1080/23746149.2016.1230476

[4] C. M. Caves, Phys. Rev. D 23, 1693 (1981).
https://doi.org/10.1103/PhysRevD.23.1693

[5] V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412, 417–419 (2001).
https://doi.org/10.1038/35086525

[6] U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys. Rev. Lett. 102, 040403 (2009).
https://doi.org/10.1103/PhysRevLett.102.040403

[7] M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Nat. Photonics 4, 357–360 (2010).
https://doi.org/10.1038/nphoton.2010.39

[8] H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, Science 337, 1514–1517 (2012).
https://doi.org/10.1126/science.1225258

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

[10] M. Paris and J. Rehacek, Quantum State Estimation, Lect. Notes Phys. (Springer Berlin Heidelberg, 2004).
https://books.google.com.au/books?id=Grr25VFtGgUC

[11] M. Hayashi, Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, 2005).
https://doi.org/10.1142/5630

[12] M. G. A. Paris, Int. J. Quantum Inf. 07, 125–137 (2009).
https://doi.org/10.1142/s0219749909004839

[13] F. Albarelli, M. Barbieri, M. Genoni, and I. Gianani, Phys. Lett. A 384, 126311 (2020).
https://doi.org/10.1016/j.physleta.2020.126311

[14] H. Yuen and M. Lax, IEEE Trans. Inf. Theory 19, 740 (1973).
https://doi.org/10.1109/TIT.1973.1055103

[15] C. Helstrom and R. Kennedy, IEEE Trans. Inf. Theory 20, 16–24 (1974).
https://doi.org/10.1109/tit.1974.1055173

[16] M. Hayashi, On simultaneous measurement of noncommutative physical values, in Development of infinite-dimensional noncommutative analysis, 1099 (RIMS Kokyuroku, Kyoto Univ., 1999) pp. 96–188.

[17] 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

[18] C. W. Helstrom, J. Stat. Phys. 1, 231–252 (1969).
https://doi.org/10.1007/bf01007479

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

[20] 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

[21] L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, Npj Quantum Inf. 7, 110 (2021).
https://doi.org/10.1038/s41534-021-00414-1

[22] S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 (1995).
https://doi.org/10.1103/PhysRevLett.74.1259

[23] R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000).
https://doi.org/10.1103/PhysRevA.61.042312

[24] A. Mansouri, R. A. Abrahao, and J. S. Lundeen, in Frontiers in Optics $+$ Laser Science 2022 (FIO, LS) (Optica Publishing Group, 2022) p. FM3B.5.
https://doi.org/10.1364/FIO.2022.FM3B.5

[25] S. Zhou, C.-L. Zou, and L. Jiang, Quantum Sci. Technol. 5, 025005 (2020).
https://doi.org/10.1088/2058-9565/ab71f8

[26] L. O. Conlon, T. Vogl, C. D. Marciniak, I. Pogorelov, S. K. Yung, F. Eilenberger, D. W. Berry, F. S. Santana, R. Blatt, T. Monz, P. K. Lam, and S. M. Assad, Nat. Phys. 19, 351–357 (2023a).
https://doi.org/10.1038/s41567-022-01875-7

[27] M. A. Ballester, Phys. Rev. A 69, 022303 (2004).
https://doi.org/10.1103/PhysRevA.69.022303

[28] Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Nat. Commun. 9, 10.1038/s41467-018-03849-x (2018).
https://doi.org/10.1038/s41467-018-03849-x

[29] Y. Yuan, Z. Hou, J.-F. Tang, A. Streltsov, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Npj Quantum Inf. 6, 10.1038/s41534-020-0280-6 (2020).
https://doi.org/10.1038/s41534-020-0280-6

[30] L. O. Conlon, F. Eilenberger, P. K. Lam, and S. M. Assad, Commun. Phys. 6, 10.1038/s42005-023-01454-z (2023b).
https://doi.org/10.1038/s42005-023-01454-z

[31] J. Pauwels, A. Pozas-Kerstjens, F. Del Santo, and N. Gisin, Phys. Rev. X 15, 021013 (2025).
https://doi.org/10.1103/PhysRevX.15.021013

[32] K. Matsumoto, J. Phys. A 35, 3111–3123 (2002).
https://doi.org/10.1088/0305-4470/35/13/307

[33] C. Helstrom, Phys. Lett. A 25, 101–102 (1967).
https://doi.org/10.1016/0375-9601(67)90366-0

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

[35] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Springer, 2011).
https://doi.org/10.1007/978-88-7642-378-9

[36] H. Nagaoka, A new approach to Cramér-Rao bounds for quantum state estimation, in Asymptotic Theory of Quantum Statistical Inference (World Scientific, 2005) p. 100–112.
https://doi.org/10.1142/9789812563071_0009

[37] H. Nagaoka, A generalization of the simultaneous diagonalization of Hermitian matrices and its relation to quantum estimation theory, in Asymptotic Theory of Quantum Statistical Inference (World Scientific, 2005) p. 133–149.
https://doi.org/10.1142/9789812563071_0012

[38] M. Hayashi, A linear programming approach to attainable Cramér-Rao type bounds, in Quantum Communication, Computing, and Measurement, edited by O. Hirota, A. S. Holevo, and C. M. Caves (Springer US, Boston, MA, 1997) pp. 99–108.
https://doi.org/10.1007/978-1-4615-5923-8_11

[39] M. Hayashi and Y. Ouyang, Quantum 7, 1094 (2023).
https://doi.org/10.22331/q-2023-08-29-1094

[40] H. Zhu, Quantum State Estimation and Symmetric Informationally Complete POMs, Phd thesis, National University of Singapore (2012).
http://scholarbank.nus.edu.sg/bitstream/handle/10635/35247/ZhuHJthesis.pdf

[41] L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, Phys. Lett. A 542, 130445 (2025).
https://doi.org/10.1016/j.physleta.2025.130445

[42] J. Kahn and M. Guţă, Commun. Math. Phys. 289, 597–652 (2009).
https://doi.org/10.1007/s00220-009-0787-3

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

[44] Y. Yang, G. Chiribella, and M. Hayashi, Commun. Math. Phys. 368, 223–293 (2019b).
https://doi.org/10.1007/s00220-019-03433-4

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

[46] L. O. Conlon, J. Suzuki, P. K. Lam, and S. Assad, The Gap Persistence Theorem Between Nagaoka-Hayashi Bound and Holevo Bound for Quantum Multiparameter Estimation, Tech. Rep. 123, 14 (IEICE, 2023).

[47] J. Zhang and J. Suzuki 10.48550/arxiv.2403.20131 (2024).
https://doi.org/10.48550/arxiv.2403.20131

[48] B. Li, L. O. Conlon, P. K. Lam, and S. M. Assad, Phys. Rev. A 108, 032605 (2023).
https://doi.org/10.1103/PhysRevA.108.032605

[49] M. Gell-Mann, Phys. Rev. 125, 1067 (1962).
https://doi.org/10.1103/PhysRev.125.1067

[50] Y. Watanabe, T. Sagawa, and M. Ueda, Phys. Rev. A 84, 042121 (2011).
https://doi.org/10.1103/PhysRevA.84.042121

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

[52] L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad 10.48550/arxiv.2208.07386 (2022).
https://doi.org/10.48550/arxiv.2208.07386

[53] A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, J. Stat. Mech.: Theory Exp. 2019 (9), 094010.
https://doi.org/10.1088/1742-5468/ab3ccb

[54] M. Tsang, F. Albarelli, and A. Datta, Phys. Rev. X 10, 031023 (2020).
https://doi.org/10.1103/PhysRevX.10.031023

[55] N. Li, C. Ferrie, J. A. Gross, A. Kalev, and C. M. Caves, Phys. Rev. Lett. 116, 180402 (2016).
https://doi.org/10.1103/PhysRevLett.116.180402

[56] H. Zhu and M. Hayashi, Phys. Rev. Lett. 120, 030404 (2018).
https://doi.org/10.1103/PhysRevLett.120.030404

[57] A. Candeloro, Z. Pazhotan, and M. G. A. Paris, Quantum Sci. Technol. 9, 045045 (2024).
https://doi.org/10.1088/2058-9565/ad7498

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

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

[60] L. O. Conlon, P. K. Lam, and S. M. Assad, Entropy 25, 1122 (2023d).
https://doi.org/10.3390/e25081122

[61] A. Fujiwara and H. Nagaoka, J. Math. Phys. 40, 4227–4239 (1999).
https://doi.org/10.1063/1.532962

[62] D. R. Cox and N. Reid, J. R. Stat. Soc., B 49, 1 (1987).
https://doi.org/10.1111/j.2517-6161.1987.tb01422.x

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

[64] R. Demkowicz-Dobrzański, W. Górecki, and M. Guţă, J. Phys. A 53, 363001 (2020).
https://doi.org/10.1088/1751-8121/ab8ef3

[65] A. Fujiwara and K. Yamagata, Ann. Stat. 51, 10.1214/23-aos2285 (2023).
https://doi.org/10.1214/23-aos2285

[66] R. A. Bertlmann and P. Krammer, J. Phys. A 41, 235303 (2008).
https://doi.org/10.1088/1751-8113/41/23/235303

[67] G. M. D’Ariano, L. Maccone, and M. G. A. Paris, J. Phys. A 34, 93–103 (2000).
https://doi.org/10.1088/0305-4470/34/1/307

[68] G. M. D Ariano, P. Perinotti, and M. F. Sacchi, J. Opt. B: Quantum Semiclass. Opt. 6, S487–S491 (2004).
https://doi.org/10.1088/1464-4266/6/6/005

[69] H. E. Haber, SciPost Phys. Lect. Notes , 21 (2021).
https://doi.org/10.21468/SciPostPhysLectNotes.21

[70] V. I. Borodulin, R. N. Rogalyov, and S. R. Slabospitskii, Core 3.2 (compendium of relations, version 3.2) (2022), arXiv:1702.08246 [hep-ph].
https://doi.org/10.48550/arXiv.1702.08246
arXiv:1702.08246

[71] A. E. Rastegin, Eur. Phys. J. D 67, 269 (2013).
https://doi.org/10.1140/epjd/e2013-40453-2

[72] G. Tóth and I. Apellaniz, J. Phys. A 47, 424006 (2014).
https://doi.org/10.1088/1751-8113/47/42/424006

Cited by

[1] Aritra Das, Simon K. Yung, Lorcán O. Conlon, Özlem Erkılıç, Angus Walsh, Yong-Su Kim, Ping K. Lam, Syed M. Assad, and Jie Zhao, "Precision bounds for characterising quantum measurements", Nature Communications 17 1, 1821 (2026).

[2] Jayanth Jayakumar, Marco Barbieri, and Magdalena Stobińska, "Measurement compatibility in multiparameter quantum interferometry", Physical Review A 112 4, 042604 (2025).

[3] Simon K. Yung, Lorcán O. Conlon, Jie Zhao, Ping Koy Lam, and Syed M. Assad, "Comparison of estimation limits for quantum two-parameter estimation", Physical Review Research 6 3, 033315 (2024).

[4] Alessandro Candeloro, Zahra Pazhotan, and Matteo G. A. Paris, "Dimension matters: precision and incompatibility in multi-parameter quantum estimation models", Quantum Science and Technology 9 4, 045045 (2024).

[5] Lorcán O. Conlon, Jun Suzuki, Ping Koy Lam, and Syed M. Assad, "The gap persistence theorem for quantum multiparameter estimation", arXiv:2208.07386, (2022).

[6] Ekaterina Moreva, Valeria Cimini, Ilaria Gianani, Ettore Bernardi, Paolo Traina, Ivo P. Degiovanni, and Marco Barbieri, "Quantum photonics sensing in biosystems", APL Photonics 10 1, 010902 (2025).

[7] Shoukang Chang, Yashu Yang, Wei Ye, Yawen Tang, Hui Cao, Huan Zhang, Zunlue Zhu, Shaoming Fei, and Xingdong Zhao, "Multiparameter quantum estimation with a uniformly accelerated Unruh-DeWitt detector", arXiv:2601.02689, (2026).

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

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