Parametric models and information geometry on W*-algebras
Tóm tắt
We introduce the notion of smooth parametric model of normal positive linear functionals on possibly infinite-dimensional
$$W^{\star }$$
-algebras generalizing the notions of parametric models used in classical and quantum information geometry. We then use the Jordan product naturally available in this context in order to define a Riemannian metric tensor on parametric models satsfying suitable regularity conditions. This Riemannian metric tensor reduces to the Fisher–Rao metric tensor, or to the Fubini-Study metric tensor, or to the Bures–Helstrom metric tensor when suitable choices for the
$$W^{\star }$$
-algebra and the models are made.
Tài liệu tham khảo
Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. Springer, New York (1988). https://doi.org/10.1007/978-1-4612-1029-0
Alfsen, E.M., Shultz, F.W.: State Spaces of Operator Algebras. Springer, New York (2001). https://doi.org/10.1007/978-1-4612-0147-2
Ali, T.S., Antoine, J.P., Gazeau, J.P.: Coherent States, Wavelets, and Their Generalizations. Springer, New York (1999). https://doi.org/10.1007/978-1-4614-8535-3
Amari, S.I.: Information Geometry and its Application. Springer, Japan (2016). https://doi.org/10.1007/978-4-431-55978-8
Amari, S.I., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, Providence (2000). https://doi.org/10.1090/mmono/191
Ashtekar, A., Schilling, T.A.: Geometrical formulation of quantum mechanics. In: Harvey, A. (ed.) On Einstein’s Path: essays in honor of Engelbert Schucking, pp. 23–65. Springer, New York (1999)
Ay, N., Jost, J., Le, H.V., Schwachhöfer, L.: Information geometry and sufficient statistics. Probab. Theory Relat. Fields 162(1), 327–364 (2015)
Ay, N., Jost, J., Le, H.V., Schwachhöfer, L.: Information Geometry (2017). https://doi.org/10.1007/978-3-319-56478-4
Ay, N., Jost, J., Le, H.V., Schwachhöfer, L.: Parametrized measure models. Bernoulli 24(3), 1692–1725 (2018)
Beltita, D., Ratiu, T.S.: Symplectic leaves in real Banach Lie-Poisson spaces. Geom. Funct. Anal. 15, 753–779 (2005). https://doi.org/10.1007/s00039-005-0524-9
Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, New York (2006). https://doi.org/10.1017/cbo9780511535048
Blackadar, B.: Operator Algebras: Theory of \(C^*\)-algebras and von Neumann Algebras. Springer, Berlin (2006)
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I, 2nd edn. Springer, Berlin (1987). https://doi.org/10.1007/978-3-662-03444-6
Cariñena, J.F., Ibort, A., Marmo, G., Morandi, G.: Geometry from Dynamics, Classical and Quantum. Springer, Dordrecht (2015). https://doi.org/10.1007/978-94-017-9220-2
Cencov, N.N.: Statistical Decision Rules and Optimal Inference. American Mathematical Society, Providence (1982). https://doi.org/10.1090/mmono/053
Choi, M.: A Schwarz inequality for positive linear maps on \(C^{\ast }\)-algebras. Ill. J. Math. 4(3), 565–574 (1974). https://doi.org/10.1215/ijm/1256051007
Ciaglia, F.M., Ibort, A., Marmo, G.: A gentle introduction to Schwinger’s formulation of quantum mechanics: the groupoid picture. Mod. Phys. Lett. A 33(20), 1850122–8 (2018). https://doi.org/10.1142/s0217732318501225
Ciaglia, F.M., Ibort, A., Marmo, G.: Schwinger’s picture of quantum mechanics I: groupoids. Int. J. Geometr. Methods Mod. Phys. 16(08), 1950119 (2019). https://doi.org/10.1142/S0219887819501196
Ciaglia, F.M., Di Cosmo, F., Ibort, A., Marmo, G.: Schwinger’s picture of quantum mechanics. Int. J. Geometr. Methods Mod. Phys. 17(04), 2050054 (2020). https://doi.org/10.1142/S0219887820500541
Ciaglia, F.M., Jost, J., Schwachhöfer, L.: Differential geometric aspects of parametric estimation theory for states on finite-dimensional C*-algebras. Entropy 22(11), 1332 (2020). https://doi.org/10.3390/e22111332
Ciaglia, F.M., Jost, J., Schwachhöfer, L.: From the Jordan product to Riemannian geometries on classical and quantum states. Entropy 22(06), 637–27 (2020). https://doi.org/10.3390/e22060637
Ciaglia, F.M., Jost, J., Schwachhöfer, L.: What can Lie algebras tell us about Jordan algebras (2021). arXiv:2112.09781 [math.DG]
Cirelli, R., Mania, A., Pizzocchero, L.: Quantum mechanics as an infinite dimensional Hamiltonian system with uncertainty structure Part I. J. Math. Phys. 31(12), 2891–2903 (1990). https://doi.org/10.1063/1.528941
D’Andrea, F., Franco, D.: On the pseudo-manifold of quantum states. Differ. Geom. Appl. 78, 101800 (2021). https://doi.org/10.1016/j.difgeo.2021.101800
Ercolessi, E., Marmo, G., Morandi, G.: From the equations of motion to the canonical commutation relations. Riv. del Nuovo Cimento 33, 401–590 (2010). https://doi.org/10.1393/ncr/i2010-10057-x
Facchi, P., Ferro, L., Marmo, G., Pascazio, S.: Defining quantumness via the Jordan product. J. Phys. A Math. Theor. (2014). https://doi.org/10.1088/1751-8113/47/3/035301
Falceto, F., Ferro, L., Ibort, M.G.: Reduction of Lie–Jordan algebras and quantum states. J. Phys. A Math. Theor. 46(1), 015201 (2013). https://doi.org/10.1088/1751-8113/46/1/015201
Fritz, T.: A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Adv. Math. 370, 107239 (2020). https://doi.org/10.1016/j.aim.2020.107239
Fritz, T., Gonda, T., Perrone, P., Rischel, E.F.: Representable markov categories and comparison of statistical experiments in categorical probability (2020). arXiv:2010.07416 [math-st]
Fritz, T., Gonda, T., Perrone, P.: De Finetti’s theorem in categorical probability (2021). arXiv:2105.02639 [math-PR]
Fujiwara, A.: Geometry of quantum information systems. In: Barndorff-Nielsen, O.E., Jensen, E.B.V. (eds.) Geometry in Present Day Science, pp. 35–48 (1999). https://doi.org/10.1142/3958
Gårding, L.: Note on continuous representations of Lie groups. Proc. Natl. Acad. Sci. USA 33(11), 331–332 (1947). https://doi.org/10.1073/pnas.33.11.331
Gibilisco, P., Isola, T.: A characterization of Wigner–Yanase skew information among statistically monotone metrics. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 4(4), 553–557 (2001). https://doi.org/10.1142/s0219025701000644
Gibilisco, P., Isola, T.: Wigner-Yanase information on quantum state space: the geometric approach. J. Math. Phys. 44(9), 3752–3762 (2003). https://doi.org/10.1063/1.1598279
Grabowski, J., Kuś, M., Marmo, G.: Symmetries, group actions, and entanglement. Open Syst. Inf. Dynam. 13(04), 343–362 (2006)
Gzyl, H., Nielsen, F.: Geometry of the probability simplex and its connection to the maximum entropy method. J. Appl. Math. Stat. Informat. 16(01), 25–35 (2020). https://doi.org/10.2478/jamsi-2020-0003
Gzyl, H., Recht, L.: A geometry on the space of probabilities I. The finite dimensional case. Rev. Mat. Iberoam. 22(02), 545–558 (2006)
Gzyl, H., Recht, L.: A geometry on the space of probabilities II. Projective spaces and exponential families. Rev. Mat. Iberoam. 22(03), 833–849 (2006)
Hasegawa, H.: Dual geometry of the Wigner–Yanase–Dyson information content. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 6(3), 413–430 (2003). https://doi.org/10.1142/S021902570300133X
Holevo, A.S.: Statistical Structure of Quantum Theory. Springer, Berlin (2001). https://doi.org/10.1007/3-540-44998-1
Holevo, A.S.: Probabilistic and statistical aspects of quantum theory. Edizioni della Normale (2011). https://doi.org/10.1007/978-88-7642-378-9
Jenčová, A.: A construction of a nonparametric quantum information manifold. J. Funct. Anal. 239(1), 1–20 (2006). https://doi.org/10.1016/j.jfa.2006.02.007
Jost, J.: Riemannian Geometry and Geometric Analysis, 7th edn. Springer, Berlin (2017)
Kibble, T.W.B.: Geometrization of quantum mechanics. Commun. Math. Phys. 65(2), 189–201 (1979)
Kirillov, A.A.: Unitary representations of nilpotent lie groups. Russ. Math. Surv. 17(4), 53–104 (1962). https://doi.org/10.1070/RM1962v017n04ABEH004118
Kirillov, A.A.: Elements of the Theory of Representations. Springer, Berlin (1976). https://doi.org/10.1007/978-3-642-66243-0
Kostant, B.: Quantization and unitary representations. In: Lectures in Modern Analysis and Applications III, Volume 170 of Lecture Notes in Mathematics. Springer, Berlin, pp. 87–208 (1970). https://doi.org/10.1007/BFb0079068
Landsman, N.P.: Mathematical Topics Between Classical and Quantum Mechanics. Springer, New York (1998). https://doi.org/10.1007/978-1-4612-1680-3
Landsman, N.P.: Foundations of Quantum Theory. From Classical Concepts to Operator Algebras. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-51777-3
Lang, S.: Fundamentals of Differential Geometry. Springer, Berlin (1999)
Liu, J., Yuan, H., Lu, X.-M., Wang, X.: Quantum Fisher information matrix and multiparameter estimation. J. Phys. A Math. Theor. 53(2), 023001–69 (2020). https://doi.org/10.1088/1751-8121/ab5d4d
Maltese, G., Niestegge, G.: A linear Radon–Nikodym type theorem for \(C^{*}\)-algebraswith applications to measure theory. Ann. della Scuola Norm. Super. di Pisa, Classe di Sci. 4, 14(2):345–354 (1987)
Man’ko, V.I., Marmo, G., Ventriglia, F., Vitale, P.: Metric on the space of quantum states from relative entropy. Tomographic reconstruction. J. Phys. A Math. Theor. 50(33), 335302 (2017). https://doi.org/10.1088/1751-8121/aa7d7d
Michor, P.W.: Manifolds of Differentiable Mappings. Shiva Publishing Limited (1980). https://doi.org/10.1007/978-3-642-11102-0_5
Naudts, J., Verbeure, A., Weder, R.: Linear response theory and the KMS condition. Commun. Math. Phys. 44, 87–99 (1975). https://doi.org/10.1007/BF01609060
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, New York (2011)
Niestegge, G.: Absolute continuity for linear forms on \(B^{*}\)-algebras and a Radon–Nikodym type theorem (quadratic version). Rend. del Circ. Mat. di Palermo 32(2), 358–376 (1983)
Paris, M.G.A.: Quantum estimation for quantum technology. Int. J. Quant. Inf. 7(1), 125–137 (2009). https://doi.org/10.1142/S0219749909004839
Parzygnat, A.J.: Inverses, disintegrations, and Bayesian inversion in quantum Markov categories (2020). arXiv:2001.08375 [quant-ph]
Perelomov, A.: Generalized Coherent States and Their Applications. Springer, Berlin (1986)
Petz, D.: Geometry of canonical correlation on the state space of a quantum system. J. Math. Phys. 35(2), 780–795 (1994). https://doi.org/10.1063/1.530611
Petz, D.: Monotone metrics on matrix spaces. Linear Algebra Appl. 244, 81–96 (1996). https://doi.org/10.1016/0024-3795(94)00211-8
Petz, D.: Quantum Information Theory and Quantum Statistics. Springer, Berlin (2007)
Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 23(5), 1543–1561 (1995)
Sakai, S.: \(C^{*}\)-Algebras and \(W^{*}\)-Algebras. Springer, Berlin (1997). https://doi.org/10.1007/978-3-642-61993-9
Souriau, J.-M.: Structure des Systèmes Dynamiques. Dunod, Paris (1970). Web source
Suzuki, J.: Information geometrical characterization of quantum statistical models in quantum estimation theory. Entropy 21(7), 703 (2019). https://doi.org/10.3390/e21070703
Takesaki, M.: Theory of Operator Algebra I. Springer, Berlin (2002)