Application of Shemesh theorem to quantum channels

TL;DR

This paper explores how the Shemesh theorem can be applied to analyze the spectral properties of quantum channels, specifically completely positive maps, to predict their long-term behavior in quantum systems.

Contribution

It introduces a novel application of the Shemesh theorem to quantum channels, linking algebraic properties of Kraus operators to spectral and asymptotic dynamics.

Findings

Connected peripheral spectrum to algebra generated by Kraus operators

Applied Shemesh and Amitsur-Levitzki theorems to quantum channel analysis

Predicted asymptotic behavior of quantum systems using algebraic structure

Abstract

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators $\mathcal{A}(A_1,\ldots A_K)$. By applying the Shemesh and Amitsur - Levitzki theorems to analyse the structure of the algebra $\mathcal{A}(A_1,\ldots A_K)$ one can predict the asymptotic dynamics for a class of operations.