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.

arXiv comment32 pages. Minor improvements. References added. Comments are welcome!

References in corpus (26)

• alpha-z-relative Renyi entropies
• Geometry of quantum systems: density states and entanglement
• Information geometry and sufficient statistics
• Metric on the space of quantum states from relative entropy. Tomographic reconstruction
• Contact manifolds and dissipation, classical and quantum
• Transition Probability (Fidelity) and Its Relatives
• Classification and characterization of quantum parametric models in quantum estimation theory
• Information geometric methods for complexity
• The Spacetime Algebra Approach to Massive Classical Electrodynamics with Magnetic Monopoles
• A gentle introduction to Schwinger's formulation of quantum mechanics: The groupoid picture
• Schwinger's Picture of Quantum Mechanics I: Groupoids
• Schwinger's Picture of Quantum Mechanics III: The statistical interpretation
• Connections and Metrics Respecting Standard Purification
• Schwinger's Picture of Quantum Mechanics II: Algebras and Observables
• Dynamical aspects in the Quantizer-Dequantizer formalism
• Reduction of Lie-Jordan Banach algebras and quantum states
• A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the $q$-$z$ Family
• Dynamical Vector Fields on the Manifold of Quantum States
• Schwinger's picture of Quantum Mechanics
• Schwinger's Picture of Quantum Mechanics IV: Composition and independence
• Manifolds of classical probability distributions and quantum density operators in infinite dimensions
• Tensorial dynamics on the space of quantum states
• Differential Calculus on Manifolds with a Boundary. Applications
• Stratified Manifold of Quantum States, actions of the complex special linear group
• Canonical divergence for measuring classical and quantum complexity
• Geometrical Structures for Classical and Quantum Probability Spaces

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• Group actions and monotone metric tensors: The qubit case
• Quantum Tomography and Schwinger's Picture of Quantum Mechanics
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