Optimal evolution models for quantum tomography

TL;DR

This paper develops optimal evolution models for quantum tomography in 2- and 3-level systems, enabling state reconstruction with a single measurement type, which could simplify experimental procedures.

Contribution

It introduces algebraic structures of quantum channels with non-degenerate generators, establishing conditions for optimal evolution models in quantum tomography.

Findings

Generators have no degenerate eigenvalues, ensuring a single observable suffices for state reconstruction.

Conditions for parameters are derived to achieve optimal evolution models.

Models have potential for experimental applications due to simplified measurement requirements.

Abstract

The research presented in this article concerns the stroboscopic approach to quantum tomography, which is an area of science where quantum Physics and linear algebra overlap. In this article we introduce the algebraic structure of the parametric-dependent quantum channels for 2-level and 3-level systems such that the generator of evolution corresponding with the Kraus operators has no degenerate eigenvalues. In such cases the index of cyclicity of the generator is equal $1$, which physically means that there exists one observable the measurement of which performed sufficient number of times at distinct instants provides enough data to reconstruct the initial density matrix and, consequently, the trajectory of the state. Necessary conditions for the parameters and relations between them are introduced. The results presented in this paper seem to have considerable potential applications in experiments due to the fact that one can perform quantum tomography by conducting only one kind of measurement. Therefore, the analyzed evolution models can be considered optimal in the context of quantum tomography. Finally, we also introduce some remarks concerning optimal evolution models in case of $n-$dimensional Hilbert space.