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The outer spectral radius and dynamics of completely positive maps
Journal article   Open access   Peer reviewed

The outer spectral radius and dynamics of completely positive maps

James E. Pascoe
Israel journal of mathematics, v 244(2), pp 945-969
01 Sep 2021
DOI: https://doi.org/10.1007/s11856-021-2198-x
url
http://arxiv.org/abs/1905.09895View
SubmittedarXiv.org - Non-exclusive license to distribute,  Open

Abstract

Mathematics Physical Sciences Science & Technology
We examine a special case of an approximation of the joint spectral radius given by Blondel and Nesterov, which we call the outer spectral radius. The outer spectral radius is given by the square root of the ordinary spectral radius of the n(2) by n(2) matrix Sigma(X) over bar (i) over bar circle times X-i. We give an analogue of the spectral radius formula for the outer spectral radius which can be used to quickly obtain the error bounds in methods based on the work of Blondel and Nesterov. The outer spectral radius is used to analyze the iterates of a completely positive map, including the special case of quantum channels. The average of the iterates of a completely positive map approach to a completely positive map where the Kraus operators span an ideal in the algebra generated by the Kraus operators of the original completely positive map. We also give an elementary treatment of Popescu's theorems on similarity to row contractions in the matrix case, describe connections to the Parrilo-Jadbabaie relaxation, and give a detailed analysis of the maximal spectrum of a completely positive map.

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