On the monotonicity of scalar curvature in classical and quantum information geometry

Journal of Mathematical Physics

Volumn 46, Issue 2, 2005, Pages

  Gibilisco, Paolo ^a   Isola, Tommaso ^a  

^a UNIVERSITY OF ROME TOR VERGATA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17044394722     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1834693     Document Type: Article

References (41)

1. Albert, P. M. and Uhlmann, A., "Stochasticity and partial order. Doubly stochastic maps and unitary mixing," in Mathematics and its Applications (Reidel, Dordrecht, 1982), Vol. 9.
2. Amari, S. and Nagaoka, H., Methods of Information Geometry (American Mathematical Society, Providence, RI, 2000).
3. Andai, A., "Monotone Riemannian metrics on density matrices with nonmonotone scalar curvature," J. Math. Phys. 44, 3675-3688 (2003).
4. Andai, A., "On the monotonicity conjecture for the curvature of the Kubo-Mori metric," math-ph/0310064vl.
5. Ando, T., "Majorization, doubly stochastic matrices, and comparison of eigenvalues," Linear Algebr. Appl. 118, 163-248 (1989).
6. Ando, T., "Majorization and inequalities in matrix theory," Linear Algebr. Appl. 199, 17-67 (1994).
7. Bhatia, R., Matrix Analysis, Graduate Texts in Mathematics, 169 (Springer, New York, 1997).
8. Brody, D. and Hughston, L., "Geometrization of statistical mechanics," Proc. R. Soc. London, Ser. A 455, 1683-1715 (1999).
9. Brody, D. and Rivier, N., "Geometrical aspects of statistical mechanics," Phys. Rev. E 51, 1006-1011 (1995).
10. Čencov, N. N., Statistical Decision Rules ana Optimal Inference (American Mathematical Society, Providence, RI, 1982) [translation from the Russian edited by Lev J. Leifman].
11. Dittmann, J., "The scalar curvature of the Bures metric on the space of density matrices," J. Geom. Phys. 31, 16-24 (1999).
12. Dittmann, J., "On the curvature of monotone metrics and a conjecture concerning the Kubo-Mori metric," Linear Algebr. Appl. 315, 83-112 (2000).
13. Dittmann, J., personal communication, 2002.
14. P-spaces," Infinite Dimen. Anal., Quantum Probab., Relat. Top. 2, 169-178 (1999).
15. 2-spaces," in Disordered and Complex Systems, AIP Conference Proceedings 553, edited by P. Sollich et al. (AIP, Melville, NY, 2001), pp. 129-139.
16. Gibilisco, P. and Isola, T., "A characterisation of Wigner-Yanase skew information among statistically monotone metrics," Infinite Dimen. Anal., Quantum Probab., Relat. Top. 4, 553-557 (2001).
17. Gibilisco, P. and Isola, T., "Wigner-Yanase information on quantum state space: The geometric approach," J. Math. Phys. 44, 3752-3762 (2003).
18. Gibilisco, P. and Isola, T., "On the characterization of paired monotone metrics," Ann. Inst. Stat. Math. 56, 369-381 (2004).
19. Gibilisco, P. and Pistone, G., "Connections on nonparametric statistical manifolds by Orlicz space geometry," Infinite Dimen. Anal., Quantum Probab., Relat. Top. 1, 325-347 (1998).
20. Gibilisco, P. and Pistone, G., "Analytical and geometrical properties of statistical connections in information geometry," in Mathematical Theory of Networks and Systems, edited by A. Beghi et al. (Il Poligrafo Padova, Italy, 1999), pp. 881-914
21. Grasselli, M. R., "Dual connections in nonparametric classical information geometry," math-ph/0104031.
22. Hasegawa, H., "Dual geometry of the Wigner-Yanase-Dyson information content," Infinite Dimen. Anal., Quantum Probab., Relat. Top. 6, 413-430 (2003).
23. Hasegawa, H. and Petz, D., "Noncommutative extension of the information geometry II," in Quantum Communications and Measurement (Plenum, New York, 1997), pp. 109-118.
24. Hiai, F., Petz, D., and Toth, G., "Curvature in the geometry of canonical correlation," Stud. Sci. Math. Hung. 32, 235-249 (1996).
25. Janke, W., Johnston, D. A., and Kenna, R., "Information geometry of the spherical model," Phys. Rev. E 67, 046106 (2003).
26. Janyszek, H., "On the geometrical structure of the generalized quantum Gibbs states," Rep. Math. Phys. 24, 11-19 (1986).
27. Janyszek, H. and Mrugala, R., "Riemannian geometry and the thermodyamics of model magnetic systems," Phys. Rev. A 39, 6515-6523 (1989).
28. Jenčová, A., "Geometry of quantum states: Dual connections and divergence functions," Rep. Math. Phys. 47, 121-138 (2001).
29. Kriegl, A. and Michor, P. W., The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, Vol. 53 (American Mathematical Society, Providence, RI, 1997).
30. Lieb, E. H., "Some convexity and subadditivity properties of entropy," Bull. Am. Math. Soc. 81, 1-14 (1975).
31. Marshall, A. W. and Olkin, I., Inequalities: Theory of Majorization and its Applications, Mathematics in Science and Engineering, 143 (Academic, New York, 1979).
32. Michor, P., Petz, D., and Andai, A., "On the curvature of a certain Riemannian space of matrices," Infinite Dimen. Anal., Quantum Probab., Relat. Top. 3, 199-212 (2000).
33. Neuwirther, M., "Submanifold geometry and hessians on the pseudo-Riemannian manifold of metrics," Acta Math. Univ. Comen. 52, 51-85 (1993).
34. Petz, D., "Geometry of canonical correlation on the state space of a quantum system," J. Math. Phys. 35, 780-795 (1994).
35. Petz, D., "Monotone metrics on matrix spaces," Linear Algebr. Appl. 244, 81-96 (1996).
36. Petz, D., "Covariance and Fisher information in quantum mechanics," J. Phys. A 35, 929-939 (2002).
37. Petz, D. and Hasegawa, H., "On the Riemannian metric of α-entropies of density matrices," Lett. Math. Phys. 38, 221-225 (1996).
38. Petz, D. and Sudár, C., "Geometry of quantum states," J. Math. Phys. 37, 2662-2673 (1996).
39. Ruppeiner, G., "Riemannian geometry approach to critical points: General theory," Phys. Rev. E 57, 5135-5145 (1998).
40. 1-PK and remarks on related inequalities," quant-ph/0404126vl.
41. Streater, R. F., Statistical Dynamics (Imperial College Press, World Scientific, Singapore, 1995).