# Tomographic Reconstruction of Quantum Metrics

## TL;DR

This paper explores the relationship between quantum metrics defined on quantum states and those defined on quantum tomograms, establishing a bijective correspondence through differential equations and illustrating with specific examples.

## Contribution

It introduces a method to relate different quantum metrics via tomographic maps, expanding understanding of quantum information geometry.

## Key findings

- Established a bijective relation between monotone quantum metrics and tomographic maps.
- Derived differential equations linking tomographic parameters to quantum metrics.
- Provided examples illustrating the connection and differences between monotone and non-monotone 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.