{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T00:53:19Z","timestamp":1776473599156,"version":"3.51.2"},"reference-count":64,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,10,23]],"date-time":"2020-10-23T00:00:00Z","timestamp":1603411200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The estimation of more than one parameter in quantum mechanics is a fundamental problem with relevant practical applications. In fact, the ultimate limits in the achievable estimation precision are ultimately linked with the non-commutativity of different observables, a peculiar property of quantum mechanics. We here consider several estimation problems for qubit systems and evaluate the corresponding quantumnessR, a measure that has been recently introduced in order to quantify how incompatible the parameters to be estimated are. In particular, R is an upper bound for the renormalized difference between the (asymptotically achievable) Holevo bound and the SLD Cram\u00e9r-Rao bound (i.e., the matrix generalization of the single-parameter quantum Cram\u00e9r-Rao bound). For all the estimation problems considered, we evaluate the quantumness R and, in order to better understand its usefulness in characterizing a multiparameter quantum statistical model, we compare it with the renormalized difference between the Holevo and the SLD-bound. Our results give evidence that R is a useful quantity to characterize multiparameter estimation problems, as for several quantum statistical model, it is equal to the difference between the bounds and, in general, their behavior qualitatively coincide. On the other hand, we also find evidence that, for certain quantum statistical models, the bound is not in tight, and thus R may overestimate the degree of quantum incompatibility between parameters.<\/jats:p>","DOI":"10.3390\/e22111197","type":"journal-article","created":{"date-parts":[[2020,10,23]],"date-time":"2020-10-23T08:59:28Z","timestamp":1603443568000},"page":"1197","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["On the Quantumness of Multiparameter Estimation Problems for Qubit Systems"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7556-9512","authenticated-orcid":false,"given":"Sholeh","family":"Razavian","sequence":"first","affiliation":[{"name":"Faculty of Physics, Azarbaijan Shahid Madani University, Tabriz 5375171379, Iran"},{"name":"Quantum Technology Lab, Dipartimento di Fisica \u201cAldo Pontremoli\u201d, Universit\u00e0 degli Studi di Milano, I-20133 Milano, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7523-7289","authenticated-orcid":false,"given":"Matteo G. A.","family":"Paris","sequence":"additional","affiliation":[{"name":"Quantum Technology Lab, Dipartimento di Fisica \u201cAldo Pontremoli\u201d, Universit\u00e0 degli Studi di Milano, I-20133 Milano, Italy"},{"name":"INFN, Sezione di Milano, I-20133 Milano, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7270-4742","authenticated-orcid":false,"given":"Marco G.","family":"Genoni","sequence":"additional","affiliation":[{"name":"Quantum Technology Lab, Dipartimento di Fisica \u201cAldo Pontremoli\u201d, Universit\u00e0 degli Studi di Milano, I-20133 Milano, Italy"},{"name":"INFN, Sezione di Milano, I-20133 Milano, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1330","DOI":"10.1126\/science.1104149","article-title":"Quantum-Enhanced Measurements: Beating the Standard Quantum Limit","volume":"306","author":"Giovannetti","year":"2004","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1142\/S0219749909004839","article-title":"Quantum estimation for quantum technology","volume":"7","author":"Paris","year":"2009","journal-title":"Int. J. Quantum Inf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1038\/nphoton.2011.35","article-title":"Advances in quantum metrology","volume":"5","author":"Giovannetti","year":"2011","journal-title":"Nat. Photonics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/bs.po.2015.02.003","article-title":"Quantum Limits in Optical Interferometry","volume":"60","author":"Wolf","year":"2015","journal-title":"Progress in Optics"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"035002","DOI":"10.1103\/RevModPhys.89.035002","article-title":"Quantum sensing","volume":"89","author":"Degen","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"035006","DOI":"10.1103\/RevModPhys.90.035006","article-title":"Quantum-enhanced measurements without entanglement","volume":"90","author":"Braun","year":"2018","journal-title":"Rev. Mod. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"035005","DOI":"10.1103\/RevModPhys.90.035005","article-title":"Quantum metrology with nonclassical states of atomic ensembles","volume":"90","author":"Smerzi","year":"2018","journal-title":"Rev. Mod. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"024014","DOI":"10.1103\/PhysRevD.94.024014","article-title":"Probing deformed quantum commutators","volume":"94","author":"Rossi","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"053805","DOI":"10.1103\/PhysRevA.93.053805","article-title":"Enhanced estimation of loss in the presence of Kerr nonlinearity","volume":"93","author":"Rossi","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"01LT01","DOI":"10.1088\/2058-9565\/aa540a","article-title":"Probing the diamagnetic term in light\u2013matter interaction","volume":"2","author":"Rossi","year":"2017","journal-title":"Quantum Sci. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"120504","DOI":"10.1103\/PhysRevLett.124.120504","article-title":"Critical Quantum Metrology with a Finite-Component Quantum Phase Transition","volume":"124","author":"Garbe","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"126311","DOI":"10.1016\/j.physleta.2020.126311","article-title":"A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging","volume":"384","author":"Albarelli","year":"2020","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"363001","DOI":"10.1088\/1751-8121\/ab8ef3","article-title":"Multi-parameter estimation beyond quantum Fisher information","volume":"53","year":"2020","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"023001","DOI":"10.1088\/1751-8121\/ab5d4d","article-title":"Quantum Fisher information matrix and multiparameter estimation","volume":"53","author":"Liu","year":"2019","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_15","first-page":"621","article-title":"Multi-parameter quantum metrology","volume":"1","author":"Szczykulska","year":"2016","journal-title":"Adv. Phys. X"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"022303","DOI":"10.1103\/PhysRevA.69.022303","article-title":"Estimation of unitary quantum operations","volume":"69","author":"Ballester","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"12","DOI":"10.2478\/qmetro-2013-0003","article-title":"Quantum estimation of a two-phase spin rotation","volume":"1","author":"Vaneph","year":"2013","journal-title":"Quantum Meas. Quantum Metrol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"012107","DOI":"10.1103\/PhysRevA.87.012107","article-title":"Optimal estimation of joint parameters in phase space","volume":"87","author":"Genoni","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"070403","DOI":"10.1103\/PhysRevLett.111.070403","article-title":"Quantum Enhanced Multiple Phase Estimation","volume":"111","author":"Humphreys","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"225301","DOI":"10.1088\/1751-8113\/46\/22\/225301","article-title":"Noise-assisted quantum transport and computation","volume":"46","author":"Tamascelli","year":"2013","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"325302","DOI":"10.1088\/1751-8113\/47\/32\/325302","article-title":"A quantum-walk-inspired adiabatic algorithm for solving graph isomorphism problems","volume":"47","author":"Tamascelli","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"042342","DOI":"10.1103\/PhysRevA.94.042342","article-title":"Gaussian systems for quantum-enhanced multiple phase estimation","volume":"94","author":"Gagatsos","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"062312","DOI":"10.1103\/PhysRevA.94.062312","article-title":"Local versus global strategies in multiparameter estimation","volume":"94","author":"Knott","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"130504","DOI":"10.1103\/PhysRevLett.119.130504","article-title":"Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases","volume":"119","author":"Ciampini","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"01LT01","DOI":"10.1088\/2058-9565\/aa9212","article-title":"Entangling measurements for multiparameter estimation with two qubits","volume":"3","author":"Roccia","year":"2018","journal-title":"Quantum Sci. Technol."},{"key":"ref_26","unstructured":"Friel, J., Palittapongarnpim, P., Albarelli, F., and Datta, A. (2020). Attainability of the Holevo-Cram\u00e9r-Rao bound for two-qubit 3D magnetometry. arXiv."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3532","DOI":"10.1038\/ncomms4532","article-title":"Joint estimation of phase and phase diffusion for quantum metrology","volume":"5","author":"Vidrighin","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"023845","DOI":"10.1103\/PhysRevA.89.023845","article-title":"Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry","volume":"89","author":"Crowley","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"032114","DOI":"10.1103\/PhysRevA.92.032114","article-title":"Weak measurements and the joint estimation of phase and phase diffusion","volume":"92","author":"Altorio","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1364\/OPTICA.5.001171","article-title":"Multiparameter approach to quantum phase estimation with limited visibility","volume":"5","author":"Roccia","year":"2018","journal-title":"Optica"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1740005","DOI":"10.1142\/S0219749917400056","article-title":"On super-resolution imaging as a multiparameter estimation problem","volume":"15","author":"Chrostowski","year":"2017","journal-title":"Int. J. Quantum Inf."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"180504","DOI":"10.1103\/PhysRevLett.121.180504","article-title":"Quantum Limited Superresolution of an Incoherent Source Pair in Three Dimensions","volume":"121","author":"Yu","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"140505","DOI":"10.1103\/PhysRevLett.122.140505","article-title":"Towards Superresolution Surface Metrology: Quantum Estimation of Angular and Axial Separations","volume":"122","author":"Napoli","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"14317","DOI":"10.1038\/srep14317","article-title":"Information complementarity: A new paradigm for decoding quantum incompatibility","volume":"5","author":"Zhu","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"123001","DOI":"10.1088\/1751-8113\/49\/12\/123001","article-title":"An invitation to quantum incompatibility","volume":"49","author":"Heinosaari","year":"2016","journal-title":"J. Phys. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0375-9601(67)90366-0","article-title":"Minimum mean-squared error of estimates in quantum statistics","volume":"25","author":"Helstrom","year":"1967","journal-title":"Phys. Lett. A"},{"key":"ref_37","unstructured":"Helstrom, C.W. (1976). Quantum Detection and Estimation Theory, Academic Press."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/0047-259X(73)90028-6","article-title":"Statistical decision theory for quantum systems","volume":"3","author":"Holevo","year":"1973","journal-title":"J. Multivar. Anal."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Holevo, A.S. (2011). Probabilistic and Statistical Aspects of Quantum Theory, Edizioni della Normale. [2nd ed.].","DOI":"10.1007\/978-88-7642-378-9"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"102101","DOI":"10.1063\/1.2988130","article-title":"Asymptotic performance of optimal state estimation in qubit system","volume":"49","author":"Hayashi","year":"2008","journal-title":"J. Math. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"052108","DOI":"10.1103\/PhysRevA.73.052108","article-title":"Local asymptotic normality for qubit states","volume":"73","author":"Kahn","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"094010","DOI":"10.1088\/1742-5468\/ab3ccb","article-title":"On quantumness in multi-parameter quantum estimation","volume":"2019","author":"Carollo","year":"2019","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"042201","DOI":"10.1063\/1.4945086","article-title":"Explicit formula for the Holevo bound for two-parameter qubit-state estimation problem","volume":"57","author":"Suzuki","year":"2016","journal-title":"J. Math. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"740","DOI":"10.1109\/TIT.1973.1055103","article-title":"Multiple-parameter quantum estimation and measurement of nonselfadjoint observables","volume":"19","author":"Yuen","year":"1973","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/BF01032091","article-title":"Generalized uncertainty relations and efficient measurements in quantum systems","volume":"26","author":"Belavkin","year":"1976","journal-title":"Theor. Math. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"042312","DOI":"10.1103\/PhysRevA.61.042312","article-title":"State estimation for large ensembles","volume":"61","author":"Gill","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1007\/s00220-009-0787-3","article-title":"Local Asymptotic Normality for Finite Dimensional Quantum Systems","volume":"289","author":"Kahn","year":"2009","journal-title":"Commun. Math. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2197","DOI":"10.1214\/13-AOS1147","article-title":"Quantum local asymptotic normality based on a new quantum likelihood ratio","volume":"41","author":"Yamagata","year":"2013","journal-title":"Ann. Stat."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s00220-019-03433-4","article-title":"Attaining the Ultimate Precision Limit in Quantum State Estimation","volume":"368","author":"Yang","year":"2019","journal-title":"Commun. Math. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"3111","DOI":"10.1088\/0305-4470\/35\/13\/307","article-title":"A new approach to the Cram\u00e9r-Rao-type bound of the pure-state model","volume":"35","author":"Matsumoto","year":"2002","journal-title":"J. Phys. A"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Albarelli, F., Tsang, M., and Datta, A. (2019). Upper bounds on the Holevo Cram\u00e9r-Rao bound for multiparameter quantum parametric and semiparametric estimation. arXiv.","DOI":"10.1103\/PhysRevLett.123.200503"},{"key":"ref_52","first-page":"031023","article-title":"Quantum Semiparametric Estimation","volume":"10","author":"Tsang","year":"2020","journal-title":"Phys. Rev. X"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-018-27362-9","article-title":"Uhlmann curvature in dissipative phase transitions","volume":"8","author":"Carollo","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"052108","DOI":"10.1103\/PhysRevA.94.052108","article-title":"Compatibility in multiparameter quantum metrology","volume":"94","author":"Ragy","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Suzuki, J. (2019). Information Geometrical Characterization of Quantum Statistical Models in Quantum Estimation Theory. Entropy, 21.","DOI":"10.3390\/e21070703"},{"key":"ref_56","unstructured":"Matsumoto, K. (1997). A new approach to the Cramer-Rao type bound of the pure state model. arXiv."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"030801","DOI":"10.1103\/PhysRevLett.116.030801","article-title":"Quantum Enhanced Estimation of a Multidimensional Field","volume":"116","author":"Baumgratz","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"2598","DOI":"10.1016\/j.physleta.2017.06.024","article-title":"A tight Cram\u00e9r\u2013Rao bound for joint parameter estimation with a pure two-mode squeezed probe","volume":"381","author":"Bradshaw","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"012106","DOI":"10.1103\/PhysRevA.97.012106","article-title":"Ultimate precision of joint quadrature parameter estimation with a Gaussian probe","volume":"97","author":"Bradshaw","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_60","unstructured":"Sidhu, J.S., and Kok, P. (2018). Quantum Fisher information for general spatial deformations of quantum emitters. arXiv."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"200503","DOI":"10.1103\/PhysRevLett.123.200503","article-title":"Evaluating the Holevo Cram\u00e9r-Rao Bound for Multiparameter Quantum Metrology","volume":"123","author":"Albarelli","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"4227","DOI":"10.1063\/1.532962","article-title":"An estimation theoretical characterization of coherent states","volume":"40","author":"Fujiwara","year":"1999","journal-title":"J. Math. Phys."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"3439","DOI":"10.1103\/PhysRevLett.72.3439","article-title":"Statistical distance and the geometry of quantum states","volume":"72","author":"Braunstein","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"244001","DOI":"10.1088\/1751-8121\/ab7f67","article-title":"Uncertainty and trade-offs in quantum multiparameter estimation","volume":"53","author":"Kull","year":"2020","journal-title":"J. Phys. A Math. Theor."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/11\/1197\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:26:59Z","timestamp":1760178419000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/11\/1197"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,23]]},"references-count":64,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["e22111197"],"URL":"https:\/\/doi.org\/10.3390\/e22111197","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,23]]}}}