{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T16:15:38Z","timestamp":1781712938300,"version":"3.54.5"},"reference-count":74,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,8]],"date-time":"2020-06-08T00:00:00Z","timestamp":1591574400000},"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 Jordan product on the self-adjoint part of a finite-dimensional     C *    -algebra    A    is shown to give rise to Riemannian metric tensors on suitable manifolds of states on    A   , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher\u2013Rao metric tensor is recovered in the Abelian case, that the Fubini\u2013Study metric tensor is recovered when we consider pure states on the algebra     B ( H )     of linear operators on a finite-dimensional Hilbert space    H   , and that the Bures\u2013Helstrom metric tensors is recovered when we consider faithful states on     B ( H )    . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on     B ( H )    , this alternative geometrical description clarifies the analogy between the Fubini\u2013Study and the Bures\u2013Helstrom metric tensor.<\/jats:p>","DOI":"10.3390\/e22060637","type":"journal-article","created":{"date-parts":[[2020,6,9]],"date-time":"2020-06-09T04:19:39Z","timestamp":1591676379000},"page":"637","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["From the Jordan Product to Riemannian Geometries on Classical and Quantum States"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8987-1181","authenticated-orcid":false,"given":"Florio M.","family":"Ciaglia","sequence":"first","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5258-6590","authenticated-orcid":false,"given":"J\u00fcrgen","family":"Jost","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4268-6923","authenticated-orcid":false,"given":"Lorenz","family":"Schwachh\u00f6fer","sequence":"additional","affiliation":[{"name":"Faculty for Mathematics, TU Dortmund University, 44221 Dortmund, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,8]]},"reference":[{"key":"ref_1","first-page":"81","article-title":"Information and accuracy attainable in the estimation of statistical parameters","volume":"37","author":"Rao","year":"1945","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1098\/rsta.1922.0009","article-title":"On the mathematical foundations of theoretical statistics","volume":"222","author":"Fisher","year":"1922","journal-title":"Philos. Trans. R. Soc. London. Ser. A"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Amari, S.I. (2016). Information Geometry and Its Application, Springer.","DOI":"10.1007\/978-4-431-55978-8"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Amari, S.I., Barndorff-Nielsen, O.E., Kass, R.E., Lauritzen, S.L., and Rao, C.R. (1987). Differential Geometry in Statistical Inference, Institute of Mathematical Statistics.","DOI":"10.1214\/lnms\/1215467056"},{"key":"ref_5","unstructured":"Amari, S.I., and Nagaoka, H. (2000). Methods of Information Geometry, American Mathematical Society."},{"key":"ref_6","unstructured":"Cencov, N.N. (1982). Statistical Decision Rules and Optimal Inference, American Mathematical Society."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"032101","DOI":"10.1063\/1.5018926","article-title":"Information geometric methods for complexity","volume":"3","author":"Felice","year":"2018","journal-title":"CHAOS"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s00440-014-0574-8","article-title":"Information geometry and sufficient statistics","volume":"162","author":"Ay","year":"2015","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ay, N., Jost, J., Le, H.V., and Schwachh\u00f6fer, L. (2017). Information Geometry, Springer International Publishing.","DOI":"10.1007\/978-3-319-56478-4"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1112\/blms\/bdw020","article-title":"Uniqueness of the Fisher\u2013Rao metric on the space of smooth densities","volume":"48","author":"Bauer","year":"2016","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_11","first-page":"135","article-title":"An extended Cencov characterization of the information metric","volume":"98","author":"Campbell","year":"1986","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0024-3795(94)00211-8","article-title":"Monotone metrics on matrix spaces","volume":"244","author":"Petz","year":"1996","journal-title":"Linear Algebra Its Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2648","DOI":"10.1007\/BF01095975","article-title":"Markov invariant geometry on state manifolds","volume":"56","author":"Morozowa","year":"1991","journal-title":"J. Sov. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"022202","DOI":"10.1063\/1.4906367","article-title":"\u03b1-z-relative Renyi entropies","volume":"56","author":"Audenaert","year":"2015","journal-title":"J. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3878","DOI":"10.3390\/e16073878","article-title":"The entropy-based quantum metric","volume":"16","author":"Balian","year":"2014","journal-title":"Entropy"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.aop.2018.05.015","article-title":"A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: The q-z family","volume":"395","author":"Ciaglia","year":"2018","journal-title":"Ann. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Felice, D., Mancini, S., and Ay, N. (2019). Canonical Divergence for Measuring Classical and Quantum Complexity. Entropy, 21.","DOI":"10.3390\/e21040435"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"335302","DOI":"10.1088\/1751-8121\/aa7d7d","article-title":"Metric on the space of quantum states from relative entropy. Tomographic reconstruction","volume":"50","author":"Marmo","year":"2017","journal-title":"J. Phys. A Math. Theorerical"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"365301","DOI":"10.1088\/1751-8121\/aa8182","article-title":"Tensorial dynamics on the space of quantum states","volume":"50","author":"Marmo","year":"2017","journal-title":"J. Phys. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.aop.2018.11.015","article-title":"Stratified manifold of quantum states, actions of the complex special linear group","volume":"400","author":"Ciaglia","year":"2019","journal-title":"Ann. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.aop.2018.09.012","article-title":"Contact manifolds and dissipation, classical and quantum","volume":"398","author":"Ciaglia","year":"2018","journal-title":"Ann. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1740003","DOI":"10.1142\/S1230161217400030","article-title":"Dynamical vector fields on the manifold of quantum states","volume":"24","author":"Ciaglia","year":"2017","journal-title":"Open Syst. Inf. Dyn."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1016\/j.aop.2017.08.025","article-title":"Dynamical aspects in the quantizer-dequantizer formalism","volume":"385","author":"Ciaglia","year":"2017","journal-title":"Ann. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1740003","DOI":"10.1142\/S0219887817400035","article-title":"Differential Calculus on Manifolds with Boundary. Applications","volume":"14","author":"Ciaglia","year":"2017","journal-title":"INternational J. Geom. Methods Mod. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1740007","DOI":"10.1142\/S021974991740007X","article-title":"Geometrical structures for classical and quantum probability spaces","volume":"15","author":"Ciaglia","year":"2017","journal-title":"Int. J. Quantum Inf."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ballico, E., Bernardi, A., Carusotto, I., Mazzucchi, S., and Moretti, V. (2019). Differential Geometry of Quantum States, Observables and Evolution. Quantum Physics and Geometry, Springer International Publishing.","DOI":"10.1007\/978-3-030-06122-7"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1063\/1.522979","article-title":"Completely positive dynamical semigroups of N-level systems","volume":"17","author":"Gorini","year":"1976","journal-title":"J. Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/BF01608499","article-title":"On the Generators of Quantum Dynamical Semigroups","volume":"48","author":"Lindblad","year":"1976","journal-title":"Commun. Math. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/s41884-019-00022-1","article-title":"Manifolds of classical probability distributions and quantum density operators in infinite dimensions","volume":"2","author":"Ciaglia","year":"2019","journal-title":"Inf. Geom."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"10217","DOI":"10.1088\/0305-4470\/38\/47\/011","article-title":"Geometry of quantum systems: Density states and entanglement","volume":"38","author":"Grabowski","year":"2005","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/s11080-006-9013-3","article-title":"Symmetries, group actions, and entanglement","volume":"13","author":"Grabowski","year":"2006","journal-title":"Open Syst. Inf. Dyn."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2050058","DOI":"10.1142\/S0219887820500589","article-title":"Schwinger\u2019s Picture of Quantum Mechanics IV: Composition and independence","volume":"17","author":"Ciaglia","year":"2020","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2050054","DOI":"10.1142\/S0219887820500541","article-title":"Schwinger\u2019s Picture of Quantum Mechanics","volume":"17","author":"Ciaglia","year":"2020","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1850122","DOI":"10.1142\/S0217732318501225","article-title":"A gentle introduction to Schwinger\u2019s formulation of quantum mechanics: The groupoid picture","volume":"33","author":"Ciaglia","year":"2018","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1950119","DOI":"10.1142\/S0219887819501196","article-title":"Schwinger\u2019s Picture of Quantum Mechanics I: Groupoids","volume":"16","author":"Ciaglia","year":"2019","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1950136","DOI":"10.1142\/S0219887819501366","article-title":"Schwinger\u2019s Picture of Quantum Mechanics II: Algebras and Observables","volume":"16","author":"Ciaglia","year":"2019","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Ciaglia, F.M., Ibort, A., and Marmo, G. (2019). Schwinger\u2019s Picture of Quantum Mechanics III: The Statistical Interpretation. Int. J. Geom. Methods Mod. Phys., 16.","DOI":"10.1142\/S0219887819501652"},{"key":"ref_38","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_39","doi-asserted-by":"crossref","unstructured":"Bengtsson, I., and \u017byczkowski, K. (2006). Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press.","DOI":"10.1017\/CBO9780511535048"},{"key":"ref_40","first-page":"73","article-title":"On the Riemannian Geometry of Finite Dimensional Mixed States","volume":"3","author":"Dittmann","year":"1993","journal-title":"Seminar Sophus Lie"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0034-4877(96)83627-5","article-title":"On the Riemannian metric on the space of density matrices","volume":"36","author":"Dittmann","year":"1995","journal-title":"Rep. Math. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1109\/TIT.1968.1054108","article-title":"The minimum variance of estimates in quantum signal detection","volume":"14","author":"Helstrom","year":"1968","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/BF01007479","article-title":"Quantum detection and estimation theory","volume":"1","author":"Helstrom","year":"1969","journal-title":"J. Stat. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/0034-4877(76)90060-4","article-title":"The transition probability in the state space of a *-algebra","volume":"9","author":"Uhlmann","year":"1976","journal-title":"Rep. Math. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/0034-4877(86)90055-8","article-title":"Parallel transport and \u201cquantum holonomy\u201d along density operators","volume":"24","author":"Uhlmann","year":"1986","journal-title":"Rep. Math. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Gielerak, R., Lukierski, J., and Popowicz, Z. (1992). The Metric of Bures and the Geometric Phase. Groups and Related Topics, Springer.","DOI":"10.1007\/978-94-011-2801-8"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1007\/s10701-009-9381-y","article-title":"Transition Probability (Fidelity) and Its Relatives","volume":"41","author":"Uhlmann","year":"2011","journal-title":"Found. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Blackadar, B. (2006). Operator Algebras: Theory of C*-algebras and von Neumann Algebras, Springer.","DOI":"10.1007\/3-540-28517-2"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Bratteli, O., and Robinson, D.W. (1987). Operator Algebras and Quantum Statistical Mechanics I, Springer. [2nd ed.].","DOI":"10.1007\/978-3-662-02520-8"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Takesaki, M. (2002). Theory of Operator Algebra I, Springer.","DOI":"10.1007\/978-3-662-10453-8"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Alfsen, E.M., and Shultz, F.W. (2001). State Spaces of Operator Algebras, Springer.","DOI":"10.1007\/978-1-4612-0147-2"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Alfsen, E.M., and Shultz, F.W. (2003). Geometry of State Spaces of Operator Algebras, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-0019-2"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"015201","DOI":"10.1088\/1751-8113\/46\/1\/015201","article-title":"Reduction of Lie-Jordan algebras and quantum states","volume":"46","author":"Falceto","year":"2013","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"29","DOI":"10.2307\/1968117","article-title":"On an algebraic generalization of the quantum mechanical formalism","volume":"35","author":"Jordan","year":"1934","journal-title":"Ann. Math."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Landsman, N.P. (1998). Mathematical Topics Between Classical and Quantum Mechanics, Springer.","DOI":"10.1007\/978-1-4612-1680-3"},{"key":"ref_56","unstructured":"Gabbay, D.M., Thagard, P., and Woods, J. (2007). Between classical and quantum. Handbook of the Philosophy of Science, North-Holland."},{"key":"ref_57","unstructured":"Upmeier, H. (1985). Symmetric Banach Manifolds and Jordan C*-algebras, Elsevier."},{"key":"ref_58","first-page":"155","article-title":"Remarks on the GNS Representation and the Geometry of Quantum States","volume":"16","author":"Marmo","year":"2009","journal-title":"Open Syst. Inf. Dyn."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Abraham, R., Marsden, J.E., and Ratiu, T. (1988). Manifolds, Tensor Analysis, and Applications, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4612-1029-0"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Jost, J. (2017). Riemannian Geometry and Geometric Analysis, Springer. [7th ed.].","DOI":"10.1007\/978-3-319-61860-9"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1142\/S0129055X94000237","article-title":"A functional representation for non-commutative C*-algebras","volume":"6","author":"Cirelli","year":"1994","journal-title":"Rev. Math. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"780","DOI":"10.1063\/1.530611","article-title":"Geometry of canonical correlation on the state space of a quantum system","volume":"35","author":"Petz","year":"1993","journal-title":"J. Math. Phys."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Besse, A.L. (1987). Einstein Manifolds, Springer.","DOI":"10.1007\/978-3-540-74311-8"},{"key":"ref_64","first-page":"401","article-title":"From the equations of motion to the canonical commutation relations","volume":"33","author":"Ercolessi","year":"2010","journal-title":"Rivista del Nuovo Cimento"},{"key":"ref_65","first-page":"329","article-title":"K\u00e4hler geometry on complex projective spaces via reduction and unfolding","volume":"39","author":"Marmo","year":"2018","journal-title":"Rendiconti di Matematica e delle sue Applicazioni"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"3246","DOI":"10.1063\/1.532884","article-title":"Connections and metrics respecting purification of quantum states","volume":"40","author":"Dittmann","year":"1999","journal-title":"J. Math. Phys."},{"key":"ref_67","first-page":"2187","article-title":"Quantum information geometry and standard purification","volume":"43","year":"1999","journal-title":"J. Math. Phys."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"3829","DOI":"10.1088\/0305-4470\/16\/16\/020","article-title":"Normal pure states of the von Neumann algebra of bounded operators as K\u00e4hler manifold","volume":"16","author":"Cirelli","year":"1983","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"2891","DOI":"10.1063\/1.528941","article-title":"Quantum mechanics as an infinite dimensional Hamiltonian system with uncertainty structure","volume":"31","author":"Cirelli","year":"1990","journal-title":"J. Math. Phys."},{"key":"ref_70","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_71","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_72","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_73","doi-asserted-by":"crossref","unstructured":"Cafaro, C., and Ali, S.A. (2007). The spacetime algebra approach to massive classical electrodynamics with magnetic monopoles. Adv. Appl. Clifford Algebr., 23\u201337.","DOI":"10.1007\/s00006-006-0014-7"},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Doran, C., and Lasenby, A. (2003). Geometric Algebra for Physicists, Cambridge University Press.","DOI":"10.1017\/CBO9780511807497"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/6\/637\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:36:53Z","timestamp":1760175413000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/6\/637"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,8]]},"references-count":74,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["e22060637"],"URL":"https:\/\/doi.org\/10.3390\/e22060637","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,8]]}}}