Metric on the space of quantum states from relative entropy. Tomographic reconstruction

TL;DR

This paper develops quantum metrics based on relative entropy, explores their properties for different quantum states, and connects them with tomographic probabilities, proposing an experimentally testable inequality.

Contribution

It introduces a family of quantum metrics derived from Tsallis entropy and links them with quantum tomography, providing new tools for quantum state analysis.

Findings

Derived quantum metrics from Tsallis entropy for full rank states.

Connected quantum metrics with tomographic Fisher-Rao metric.

Proposed a new inequality for spin-1/2 projections to be tested experimentally.

Abstract

In the framework of quantum information geometry, we derive, from quantum relative Tsallis entropy, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential calculus. The cases N = 2, N = 3 are discussed in detail and notable limits are analyzed. The radial limit procedure has been used to recover quantum metrics for lower rank states, such as pure states. By using the tomographic picture of quantum mechanics we have obtained the Fisher- Rao metric for the space of quantum tomograms and derived a reconstruction formula of the quantum metric of density states out of the tomographic one. A new inequality obtained for probabilities of three spin-1/2 projections in three perpendicular directions is proposed to be checked in experiments with superconducting circuits.