Duality, monotonicity and the Wigner-Yanase-Dyson metrics

Infinite Dimensional Analysis, Quantum Probability and Related Topics

Volumn 7, Issue 2, 2004, Pages 215-232

  Grasselli, M R ^a  

^a MCMASTER UNIVERSITY   (Canada)

Author keywords

Ameri duality; Monotone metrics; Quantum information geometry

Indexed keywords

EID: 7444268507     PISSN: 02190257     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021902570400161X     Document Type: Article

References (36)

1. S.-I. Amari, Differential-Geometrical Methods in Statistics (Springer-Verlag, 1985).
2. S.-I. Amari and H. Nagaoka, Methods of Information Geometry (Amer. Math. Soc., 2000). Translated from the 1993 Japanese version by Daishi Harada.
3. R. Balian, Y. Alhassid and H. Reinhardt, Dissipation in many-body systems: A geometric approach based on information theory, Phys. Rep. 131 (1986) 1-146.
4. N. N. Čencov, Statistical Decision Rules and Optimal Inference (American Mathematical Society, 1982). Translation from the Russian edited by L. J. Leifman.
5. R. H. Fowler, Statistical Mechanics (Cambridge University Press, 1980). The theory of the properties of matter in equilibrium, Reprint of the second edition of 1936.
6. R. Frost, Stopping by woods on a snowy evening, 1923.
7. p-spaces, Infin. Dim. Anal. Quantum Probab. Rel. Topics. 2 (1999) 169-178.
8. 2-spaces, in Disordered and Complex Systems, eds. P. Sollich et al., AIP Conference Proceedings 553 (Amer. Inst. Physics, 2001), pp. 129-139.
9. P. Gibilisco and G. Pistone, Connections on non-parametric statistical manifolds by Orlicz space geometry, Infin. Dim. Anal. Quantum Probab. Rel. Topics 1 (1998) 325-347.
10. M. R. Grasselli, Classical and quantum information geometry, PhD thesis, King's College London, 2001.
11. M. R. Grasselli, Dual connections in nonparametric classical information geometry, submitted to Ann. Statist. Math., 2004.
12. M. R. Grasselli and R. F. Streater, The quantum information manifold for ∈-bounded forms, Rep. Math. Phys. 46 (2000) 325-335.
13. M. R. Grasselli and R. F. Streater, On the uniqueness of the Chentsov metric in quantum information geometry, Infin. Dim. Anal. Quantum Probab. Rel. Topics. 4 (2001) 173-182.
14. H. Hasegawa, α-divergence of the noncommutative information geometry, in Proc. of the XXV Symp. on Mathematical Physics (Toruń, 1992), Vol. 33, pp. 87-93, 1993.
15. H. Hasegawa, Noncommutative extension of the information geometry, in Quantum Communications and Measurement (Nottingham, 1994) (Plenum, 1995), pp. 327-337.
16. H. Hasegawa, Exponential and mixture families in quantum statistics: Dual structure and unbiased parameter estimation, Rep. Math. Phys. 39 (1997) 49-68.
17. H. Hasegawa and D. Petz, Noncommutative extension of the information geometry II, in Quantum Communications and Measurement (Plenum, 1997), pp. 109-118.
18. E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106 (1957) 620-630.
19. E. T. Jaynes, Information theory and statistical mechanics. II, Phys. Rev. 108 (1957) 171-190.
20. A. Jenčová, Dualistic properties of the manifold of quantum states, in Disordered and Complex Systems (London, 2000), AIP Conf. Proc., Vol. 553 (Amer. Inst. Phys., 2001), pp. 147-152.
21. A. Jenčová, Geometry of quantum states: Dual connections and divergence functions, Rep. Math. Phys. 47 (2001) 121-138.
22. A. Lesniewski and M. B. Ruskai, Monotone Riemannian metrics and relative entropy on noncommutative probability spaces, J. Math. Phys. 40 (1999) 5702-5724.
23. M. K. Murray and J. W. Rice, Differential Geometry and Statistics (Chapman & Hall, 1993).
24. H. Nagaoka, Differential geometrical aspects of quantum state estimation and relative entropy, in Quantum Communications and Measurement (Nottingham, 1994) (Plenum, 1995), pp. 449-452.
25. D. Petz, Monotone metrics on matrix spaces, Linear Alg. Appl. 244 (1996) 81-96.
26. D. Petz, Covariance and Fisher information in quantum mechanics, J. Phys. A35 (2002) 929-939.
27. D. Petz and H. Hasegawa, On the Riemannian metric of α-entropies of density matrices, Lett. Math. Phys. 38 (1996) 221-225.
28. G. Pistone and C. Sempi, An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one, Ann. Statist. 23 (1995) 1543-1561.
29. R. F. Streater, Statistical Dynamics (Imperial College Press, 1995).
30. R. F. Streater, Information geometry and reduced quantum description, in Proc. of the XXVIII Symp. on Mathematical Physics (Toruń, 1995), Vol. 38 (Plenum, 1996), pp. 419-436.
31. R. F. Streater, Statistical dynamics and information geometry, in Geometry and Nature (Madeira, 1995) (Amer. Math. Soc., 1997), pp. 117-131.
32. R. F. Streater, The analytic quantum information manifold, in Stochastic Processes, Physics and Geometry: New Interplays, II (Leipzig, 1999) (Amer. Math. Soc., 2000), pp. 603-611.
33. R. F. Streater, The information manifold for relatively bounded potentials, Tr. Mat. Inst. Stek., 228 Probl. Sovrem. Mat. Fiz., 217-235, 2000, On the Ninetieth Anniversary of Academician Nikolaǐ Nikolaevich Bogolyubov.
34. R. F. Streater, Dual structures on a quantum information manifold, presented at the Conference on Information Geometry and its Applications, Pescara, July 1-5, 2002.
35. A. Uhlmann, Density operators as an arena for differential geometry, in Proc. of the XXV Symp. on Mathematical Physics (Toruń, 1992), Vol. 33, pp. 253-263, 1993.
36. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, 1996). Translated from the German and with a preface by Robert T. Beyer.