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. Here we determine its scalar curvature and Ricci tensor and prove a lower bound for the curvature on the submanifold of trace one matrices. This bound is achieved for the maximally mixed state, a further hint for the quantum statistical meaning of the scalar curvature.

Latex, 9 pages

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• A priori probability that a qubit-qutrit pair is separable
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• Measure of the density of quantum states in information geometry and its application in the quantum multi-parameter estimation
• Bures Geometry of the Three-Level Quantum Systems. I
• Quantum and Fisher Information from the Husimi and Related Distributions