Abstract
A natural Riemannian geometry is defined on the state space of a finite quantum system by means of the Bogoliubov scalar product which is infinitesimally induced by the (nonsymmetric) relative entropy functional. The basic geometrical quantities, including sectional curvatures, are computed for a two-level quantum system. It is found that the real density matrices form a totally geodesic submanifold and the von Neumann entropy is a monotone function of the scalar curvature. Furthermore, we establish information inequalities extending the Cramér-Rao inequality of classical statistics. These are based on a very general new form of the logarithmic derivative.
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This work was supported by the Hungarian National Foundation for Scientific Research, grant No. 1900. Authors' e-mail addresses are: H1128PET@ella.hu and TOTH@zodiac.rutgers.edu.
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Petz, D., Toth, G. The Bogoliubov inner product in quantum statistics. Lett Math Phys 27, 205–216 (1993). https://doi.org/10.1007/BF00739578
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DOI: https://doi.org/10.1007/BF00739578