1School of Data Science, The Chinese University of Hong Kong, Shenzhen, Longgang District, Shenzhen, 518172, China
2International Quantum Academy (SIQA), Futian District, Shenzhen 518048, China
3Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
4Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom
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Abstract
In the quest to unlock the maximum potential of quantum sensors, it is of paramount importance to have practical measurement strategies that can estimate incompatible parameters with best precisions possible. However, it is still not known how to find practical measurements with optimal precisions, even for uncorrelated measurements over probe states. Here, we give a concrete way to find uncorrelated measurement strategies with optimal precisions. We solve this fundamental problem by introducing a framework of conic programming that unifies the theory of precision bounds for multiparameter estimates for uncorrelated and correlated measurement strategies under a common umbrella. Namely, we give precision bounds that arise from linear programs on various cones defined on a tensor product space of matrices, including a particular cone of separable matrices. Subsequently, our theory allows us to develop an efficient algorithm that calculates both upper and lower bounds for the ultimate precision bound for uncorrelated measurement strategies, where these bounds can be tight. In particular, the uncorrelated measurement strategy that arises from our theory saturates the upper bound to the ultimate precision bound. Also, we show numerically that there is a strict gap between the previous efficiently computable bounds and the ultimate precision bound.
Featured image: Different Cramer-Rao bounds for multiparameter quantum metrology involve optimizations of the same objective function over different cones, with the cones having a nested structure. The optimization variable is an operator on a tensor product space, with one part that corresponds to the number of parameters to be estimated, and another part that corresponds to the space for the probe state.
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[1] C. Helstrom, Minimum mean-squared error of estimates in quantum statistics, Physics Letters A 25, 101 (1967).
https://doi.org/10.1016/0375-9601(67)90366-0
[2] C. W. Helstrom, Quantum detection and estimation theory (Academic press, 1976).
[3] A. S. Holevo, Probabilistic and statistical aspects of quantum theory (Edizioni della Normale, 2011).
https://doi.org/10.1007/978-88-7642-378-9
[4] H. Nagaoka, A new approach to Cramér-Rao bounds for quantum state estimation, IEICE Tech Report IT 89-42, 9 (1989), (Reprinted in hayashi).
https://doi.org/10.1142/9789812563071_0009
[5] M. Hayashi and K. Matsumoto, Asymptotic performance of optimal state estimation in qubit system, Journal of Mathematical Physics 49, 102101 (2008).
https://doi.org/10.1063/1.2988130
[6] R. Demkowicz-Dobrzański, W. Górecki, and M. Guţă, Multi-parameter estimation beyond quantum fisher information, Journal of Physics A: Mathematical and Theoretical 53, 363001 (2020).
https://doi.org/10.1088/1751-8121/ab8ef3
[7] J. S. Sidhu and P. Kok, Geometric perspective on quantum parameter estimation, AVS Quantum Science 2, 014701 (2020).
https://doi.org/10.1116/1.5119961
[8] F. Albarelli, J. F. Friel, and A. Datta, Evaluating the Holevo Cramér-Rao bound for multiparameter quantum metrology, Phys. Rev. Lett. 123, 200503 (2019).
https://doi.org/10.1103/PhysRevLett.123.200503
[9] J. S. Sidhu, Y. Ouyang, E. T. Campbell, and P. Kok, Tight bounds on the simultaneous estimation of incompatible parameters, Phys. Rev. X 11, 011028 (2021).
https://doi.org/10.1103/PhysRevX.11.011028
[10] 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: Selected Papers, edited by M. Hayashi (World Scientific, 2005) pp. 133–149.
https://doi.org/10.1142/9789812563071_0012
[11] M. Hayashi, On simultaneous measurement of noncommutative observables. in development of infinite-dimensional non-commutative anaysis, in Surikaisekikenkyusho (RIMS), Kokyuroku No. 1099, In Japanese (Kyoto Univ., 1999) pp. 96–188.
https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1099-8.pdf
[12] L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, Efficient computation of the Nagaoka–Hayashi bound for multiparameter estimation with separable measurements, npj Quantum Information 7, 1 (2021).
https://doi.org/10.1038/s41534-021-00414-1
[13] M. Hayashi, A linear programming approach to attainable cramér-rao type bounds and randomness condition (1997a), arXiv:quant-ph/9704044 [quant-ph].
arXiv:quant-ph/9704044
[14] L. Gurvits, Classical deterministic complexity of Edmonds’ problem and quantum entanglement, STOC ’03, 10 (2003).
https://doi.org/10.1145/780542.780545
[15] D. Bruß, Characterizing entanglement, Journal of Mathematical Physics 43, 4237 (2002).
https://doi.org/10.1063/1.1494474
[16] R. Uola, T. Kraft, J. Shang, X.-D. Yu, and O. Gühne, Quantifying quantum resources with conic programming, Phys. Rev. Lett. 122, 130404 (2019).
https://doi.org/10.1103/PhysRevLett.122.130404
[17] R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso, Operational advantage of quantum resources in subchannel discrimination, Phys. Rev. Lett. 122, 140402 (2019).
https://doi.org/10.1103/PhysRevLett.122.140402
[18] R. Takagi and B. Regula, General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks, Phys. Rev. X 9, 031053 (2019).
https://doi.org/10.1103/PhysRevX.9.031053
[19] S.-I. Amari and H. Nagaoka, Methods of information geometry (American Mathematical Soc., 2007).
https://doi.org/10.1090/mmono/191
[20] H. Masahito, ed., Asymptotic theory of quantum statistical inference: selected papers (World Scientific, 2005).
https://doi.org/10.1142/5630
[21] D. Petz, Quantum information theory and quantum statistics (Springer Science & Business Media, 2007).
https://doi.org/10.1007/978-3-540-74636-2
[22] J. Suzuki, Y. Yang, and M. Hayashi, Quantum state estimation with nuisance parameters, Journal of Physics A: Mathematical and Theoretical 53, 453001 (2020).
https://doi.org/10.1088/1751-8121/ab8b78
[23] A. Fujiwara and H. Nagaoka, Quantum fisher metric and estimation for pure state models, Physics Letters A 201, 119 (1995).
https://doi.org/10.1016/0375-9601(95)00269-9
[24] A. Fujiwara and H. Nagaoka, An estimation theoretical characterization of coherent states, Journal of Mathematical Physics 40, 4227 (1999).
https://doi.org/10.1063/1.532962
[25] M. Hayashi, Quantum information theory: Mathematical Foundation (Springer, 2016) graduate Texts in Physics, First edition was published from Springer in 2006.
https://doi.org/10.1007/978-3-662-49725-8
[26] M. Hayashi and K. Matsumoto, Statistical model with measurement degree of freedom and quantum physics, in Asymptotic Theory Of Quantum Statistical Inference: Selected Papers, edited by M. Hayashi (World Scientific, 2005) pp. 162–169, originally published in Japanese in Surikaiseki Kenkyusho Kokyuroku No. 1055, 1998.
https://doi.org/10.1142/9789812563071_0014%5D
[27] R. D. Gill and S. Massar, State estimation for large ensembles, Phys. Rev. A 61, 042312 (2000).
https://doi.org/10.1103/PhysRevA.61.042312
[28] M. Hayashi, Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation, Commun. Math. Phys. 304, 689 (2011).
https://doi.org/10.1007/s00220-011-1239-4
[29] Y. Yang, G. Chiribella, and M. Hayashi, Attaining the ultimate precision limit in quantum state estimation, Communications in Mathematical Physics 36 8, 223 (2019).
https://doi.org/10.1007/s00220-019-03433-4
[30] S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Compatibility in multiparameter quantum metrology, Phys. Rev. A 94, 052108 (2016).
https://doi.org/10.1103/PhysRevA.94.052108
[31] Review of Holevo’s research in the 1980s, private paper submitted for review, and to appear on arxiv soon (2023).
[32] K. Matsumoto, A new approach to the Cramér-Rao-type bound of the pure-state model, Journal of Physics A: Mathematical and General 35, 3111 (2002).
https://doi.org/10.1088/0305-4470/35/13/307
[33] M. Guţă and J. Kahn, Local asymptotic normality for qubit states, Physical Review A 73, 052108 (2006).
https://doi.org/10.1103/PHYSREVA.73.0
[34] J. Kahn and M. Guţă, Local asymptotic normality for finite dimensional quantum systems, Communications in Mathematical Physics 289, 597 (2009).
https://doi.org/10.1007/s00220-009-0787-3
[35] K. Yamagata, A. Fujiwara, and R. D. Gill, Quantum local asymptotic normality based on a new quantum likelihood ratio, The Annals of Statistics 41, 2197 (2013).
https://doi.org/10.1214/13-AOS1147
[36] X.-M. Lu and X. Wang, Incorporating heisenberg’s uncertainty principle into quantum multiparameter estimation, Phys. Rev. Lett. 126, 120503 (2021).
https://doi.org/10.1103/PhysRevLett.126.120503
[37] M. Hayashi, A linear programming approach to attainable Cramer-Rao type bound, in Quantum Communication, Computing, and Measurement, edited by O. Hirota, A. S. Holevo, and C. M. Caves (Plenum, New York, 1997).
https://doi.org/10.48550/arXiv.quant-ph/9704044
arXiv:quant-ph/9704044
[38] M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Physics Letters A 223, 1 (1996).
https://doi.org/10.1016/S0375-9601(96)00706-2
[39] B. M. Terhal, A family of indecomposable positive linear maps based on entangled quantum states, Linear Algebra and its Applications 323, 61 (2001).
https://doi.org/10.1016/S0024-3795(00)00251-2
[40] M. Lewenstein, B. Kraus, P. Horodecki, and J. I. Cirac, Characterization of separable states and entanglement witnesses, Phys. Rev. A 63, 044304 (2001).
https://doi.org/10.1103/PhysRevA.63.044304
[41] M. Navascués, Pure state estimation and the characterization of entanglement, Phys. Rev. Lett. 100, 070503 (2008).
https://doi.org/10.1103/PhysRevLett.100.070503
[42] R. Demkowicz-Dobrzański, Optimal phase estimation with arbitrary a priori knowledge, Phys. Rev. A 83, 061802 (2011).
https://doi.org/10.1103/PhysRevA.83.061802
[43] E. Chitambar, I. George, B. Doolittle, and M. Junge, The communication value of a quantum channel, IEEE Transactions on Information Theory 69, 1660 (2023).
https://doi.org/10.1109/TIT.2022.3218540
[44] K. Fujisawa, M. Kojima, and K. Nakata, Exploiting sparsity in primal-dual interior-point methods for semidefinite programming, Mathematical Programming 79, 235 (1997).
https://doi.org/10.1007/BF02614319
[45] A. Ambainis and J. Emerson, Quantum t-designs: t-wise independence in the quantum world, in Twenty-Second Annual IEEE Conference on Computational Complexity (CCC’07) (2007) pp. 129–140, quant-ph/0701126.
https://doi.org/10.1109/CCC.2007.26
arXiv:quant-ph/0701126
[46] M. Hayashi, Group Representation for Quantum Theory (Springer, 2017).
https://doi.org/10.1007/978-3-319-44906-7
[47] C. An and Y. Xiao, Numerical construction of spherical $t$-designs by Barzilai-Borwein method, Applied Numerical Mathematics 150, 295 (2020).
https://doi.org/10.1016/j.apnum.2019.10.008
[48] P. Delsarte, J. Goethals, and J. Seidel, Spherical codes and designs, Geom Dedicata 6, 363 (1977).
https://doi.org/10.1007/BF03187604
[49] M. S. Baladram, On explicit construction of simplex $t$-designs, Interdisciplinary Information Sciences 24, 181 (2018).
https://doi.org/10.4036/iis.2018.S.02
[50] E. Bannai, E. Bannai, S. Suda, and H. Tanaka, On relative $ t $-designs in polynomial association schemes, Electronic Journal of Combinatorics 22, 1392 (2015).
https://doi.org/10.37236/4889
[51] E. Bannai and E. Bannai, A survey on spherical designs and algebraic combinatorics on spheres, European Journal of Combinatorics 30, 1392 (2009).
https://doi.org/10.1016/j.ejc.2008.11.007
[52] Y. Ouyang, Computing spectral bounds of the heisenberg ferromagnet from geometric considerations, Journal of Mathematical Physics 60, 071901 (2019).
https://doi.org/10.1063/1.5084136
[53] G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976).
https://doi.org/10.1007/BF01608499
[54] Y. Yang, G. Chiribella, and M. Hayashi, Quantum stopwatch: how to store time in a quantum memory, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, 20170773 (2018).
https://doi.org/10.1098/rspa.2017.0773
[55] G. Tóth and I. Apellaniz, Quantum metrology from a quantum information science perspective, Journal of Physics A: Mathematical and Theoretical 47, 424006 (2014).
https://doi.org/10.1088/1751-8113/47/42/424006
[56] A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Complete family of separability criteria, Phys. Rev. A 69, 022308 (2004).
https://doi.org/10.1103/PhysRevA.69.022308
[57] M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, M. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, Joint estimation of phase and phase diffusion for quantum metrology., Nat Commun 5, 3532 (2014).
https://doi.org/10.1038/ncomms4532
[58] H. Arai and M. Hayashi, Pseudo standard entanglement structure cannot be distinguished from standard entanglement structure, New Journal of Physics 25, 023009 (2023).
https://doi.org/10.1088/1367-2630/acb565
[59] R. M. Van Slyke and R. J.-B. Wets, A duality theory for abstract mathematical programs with applications to optimal control theory, Journal of Mathematical Analysis and Applications 22, 679 (1968).
https://doi.org/10.1016/0022-247X(68)90206-0
Cited by
[1] Lorcán O. Conlon, Jun Suzuki, Ping Koy Lam, and Syed M. Assad, “The gap persistence theorem for quantum multiparameter estimation”, arXiv:2208.07386, (2022).
[2] Yingkai Ouyang and Gavin K. Brennen, “Quantum error correction on symmetric quantum sensors”, arXiv:2212.06285, (2022).
[3] Yingkai Ouyang and Narayanan Rengaswamy, “Describing quantum metrology with erasure errors using weight distributions of classical codes”, Physical Review A 107 2, 022620 (2023).
[4] Lorcán O. Conlon, Ping Koy Lam, and Syed M. Assad, “Multiparameter Estimation with Two-Qubit Probes in Noisy Channels”, Entropy 25 8, 1122 (2023).
The above citations are from SAO/NASA ADS (last updated successfully 2023-08-30 12:00:33). The list may be incomplete as not all publishers provide suitable and complete citation data.
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This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
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