From the Jordan Product to Riemannian Geometries on Classical and Quantum States

Area of Science:

• Mathematical Physics
• Differential Geometry
• Quantum Information Theory

Background:

• The Jordan product, defined on the self-adjoint part of C*-algebras, has connections to geometric structures.
• Understanding the geometry of state spaces is crucial in various fields, including quantum mechanics and information theory.

Purpose of the Study:

• To investigate the Riemannian geometry induced by the Jordan product on manifolds of states of finite-dimensional C*-algebras.
• To compute key geometric quantities such as covariant derivative, geodesics, Riemann tensor, and sectional curvature.
• To establish connections between these geometric structures and well-known metric tensors.

Main Methods:

• The Jordan product is utilized to define Riemannian metric tensors on state manifolds.
• Explicit computations of covariant derivatives, geodesics, Riemann tensors, and sectional curvatures are performed.
• The Gelfand-Naimark-Segal (GNS) construction is employed for an alternative derivation of the metric tensors.

Main Results:

• The Jordan product generates Riemannian metric tensors on suitable manifolds of states.
• The Fisher-Rao metric tensor is recovered in the Abelian case.
• The Fubini-Study and Bures-Helstrom metric tensors are recovered for pure and faithful states on B(H), respectively.
• An alternative derivation using the GNS construction is presented, clarifying analogies between metric tensors.

Conclusions:

• The Jordan product provides a unified framework for generating various Riemannian metric tensors on state manifolds.
• The geometric interpretation via the GNS construction offers deeper insights into the relationships between different metric tensors.
• This work bridges abstract algebraic structures with concrete geometric concepts in quantum state spaces.