Tomographic Reconstruction of Quantum Metrics

Abstract

In the framework of quantum information geometry we investigate the relationship between monotone metric tensors uniquely defined on the space of quantum tomograms, once the tomographic scheme chosen, and monotone quantum metrics on the space of quantum states, classified by operator monotone functions, according to Petz classification theorem. We show that different metrics can be related through a change of the tomographic map and prove that there exists a bijective relation between monotone quantum metrics associated with different operator monotone functions. Such bijective relation is uniquely defined in terms of solutions of a first order second degree differential equation for the parameters of the involved tomographic maps. We first exhibit an example of a non-linear tomographic map which connects a monotone metric with a new one which is not monotone. Then we provide a second example where two monotone metrics are uniquely related through their tomographic parameters.

v1:19 pages, 2 figures. Abstract modified, references added, minor corrections. v2:Revised versione accepted for publication

References in corpus (6)

• Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics
• Geometry of quantum systems: density states and entanglement
• Star products, duality and double Lie algebras
• Metric on the space of quantum states from relative entropy. Tomographic reconstruction
• Duality symmetry for star products
• A pedagogical presentation of a $C^{\star }$-algebraic approach to quantum tomography

Cited by in corpus (6)

• Fisher Metric, Geometric Entanglement and Spin Networks
• A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the $q$-$z$ Family
• Lie groupoids in information geometry
• Stratified Manifold of Quantum States, actions of the complex special linear group
• Information geometry on groupoids: the case of singular metrics
• Geometry from divergence functions and complex structures