From the Jordan product to Riemannian geometries on classical and quantum states

Abstract

The Jordan product on the self-adjoint part of a finite-dimensional $C^{\ast }$-algebra $\mathcal{A}$ is shown to give rise to Riemannian metric tensors on suitable manifolds of states on $\mathcal{A}$, and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher--Rao metric tensor is recovered in the Abelian case, that the Fubini--Study metric tensor is recovered when we consider pure states on the algebra $\mathcal{B}(\mathcal{H})$ of linear operators on a finite-dimensional Hilbert space $\mathcal{H}$, and that the Bures--Helstrom metric tensors is recovered when we consider faithful states on $\mathcal{B}(\mathcal{H})$. Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on $\mathcal{B}(\mathcal{H})$, this alternative geometrical description clarifies the analogy between the Fubini--Study and the Bures--Helstrom metric tensor.

32 pages. Minor improvements. References added. Comments are welcome!

References in corpus (4)

• Geometry of quantum systems: density states and entanglement
• The Spacetime Algebra Approach to Massive Classical Electrodynamics with Magnetic Monopoles
• Schwinger's Picture of Quantum Mechanics IV: Composition and independence
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