The scalar curvature of the Bures metric on the space of density matrices
Abstract
The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Therefore, and also for mathematical reasons, it is natural to ask for curvature properties of this Riemannian metric. Here we determine its scalar curvature and Ricci tensor and prove a lower bound for the curvature on the submanifold of trace1 matrices. This bound is achieved for the maximally mixed state, a further hint for the statistical meaning of the scalar curvature.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 August 1999
 DOI:
 10.1016/S03930440(98)000680
 arXiv:
 arXiv:quantph/9810012
 Bibcode:
 1999JGP....31...16D
 Keywords:

 Quantum Physics;
 Mathematical Physics;
 Mathematics  Differential Geometry
 EPrint:
 Latex, 9 pages