Abstract
The Riemannian metric induced by quantum α-entropies is proven to be monotone under stochastic mappings on the set of density matrices. The length of tangent vectors is essentially the Wigner-Yanase-Dyson skew information in this setting.
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BalianR., AlhassidY. and ReinhardtH.: Dissipation in many-body systems: A geometric approach based on information theory, Phys. Rep. 131 (1986), 1–146.
BraunsteinS. L. and CavesC. M.: Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72 (1994), 3439–3443.
ČencovN. N.: Statistical Decision Rules and Optimal Inferences, Amer. Math. Soc., Providence, 1982.
Dittmann, J., On the Riemannian metric on the space of density matrices, Universität Leipzig, NTZ-Preprint No. 12/1995.
HasegawaH.: α-divergence of the non-commutative information geometry, Rep. Math. Phys. 33 (1993), 87–93.
HasegawaH.: Non-commutative extension of the information geometry, in: V. P.Belavkin, O.Hirota and R. L.Hudson (eds), Proc. Intern. Workshop on Quantum Communications and Measurement, Nottingham 1994, Plenum, New York, 1995.
Hiai, F., Petz, D. and Toth, G.: Curvature in the geometry of canonical correlation, to be published in Studia Sci. Math.
IngardenR. S., JanyszekH., KossakowskiA. and KawaguchiT.: Information geometry and quantum statistical systems, Tensor N.S. 37 (1982), 105–111.
LiebE. H.: Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. in Math. 11 (1973), 267–288.
MorozovaE. A. and ČencovN. N.: Markovian invariant geometry on manifolds of states (Russian), Itogi Nauki i Tehniki 36 (1990), 69–102.
NagaokaH.: A new approach to Cramér-Rao bounds for quantum state estimation, IEICE Technical Report, 89 (1989), No. 228, IT89-42, 9–14.
OhyaM. and PetzD.: Quantum Entropy and Its Use, Springer-Verlag, Heildelberg, 1993.
PetzD.: Geometry of canonical correlation on the state space of a quantum system, J. Math. Phys. 35 (1994), 780–795.
Petz, D.: Monotone metrics on matrix spaces, to appear in Linear Algebra Appl.
Petz, D. and Sudár, Cs.: Geometries of quantum states, Preprint ESI 204, Vienna, 1995.
PetzD. and TothG.: The Bogoliubov inner product in quantum statistics, Lett. Math. Phys. 27 (1993), 205–216.
Sudár, Cs.: Radial extension of monotone Riemannian metrics on density matrices, to be published in: Publ. Math. Debrecen.
UhlmannA.: The metric Bures and the geometric phase, in R.Gielerak et al. (ed.), Groups and Related Topics, Kluwer Acad. Publ., Dordrecht, 1992, pp. 267–274.
UhlmannA.: Density operators as an arena for differential geometry, Rep. Math. Phys. 33 (1993), 253–263.
WignerE. P. and YanaseM. M.: Information content of distributions, Proc. Nat. Acad. Sci. USA 49 (1963), 910–918.
YuenH. P. H. and LaxM.: Multiple parameter quantum estimation and measurement of non-selfadjoint observables, IEEE Trans. IT-19 (1973), 740–750.
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Supported by the Hungarian National Foundation for Scientific Research, grant No. T-016924.
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Petz, D., Hasegawa, H. On the Riemannian metric of α-entropies of density matrices. Lett Math Phys 38, 221–225 (1996). https://doi.org/10.1007/BF00398324
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DOI: https://doi.org/10.1007/BF00398324