Application of Shemesh theorem to quantum channels

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 A(A 1 , . . . A K ). By applying the Shemesh and Amitsur -Levitzki theorems to analyse the structure of the algebra A(A 1 , . . . A K ) one can predict the asymptotic dynamics for a class of operations.

Statistics of projective measurement on a quantum probe as a witness of noncommutativity of algebra of a probed system

“…Both operations belong to the class of quantum operation called random unitary channels, which are completely positive, trace preserving, and unital maps (the maps that preserve the identity.) The spectral structure and analysis, and fixed point properties for the maps in this class were examined in detail in literature, see e.g., [39][40][41][42]. When KC given in Eq.…”

Uma conjectura: o teorema de Shemesh e a capacidade erro-zero de canais quânticos com dois operadores de kraus

“…Recentemente, em 2018, o mesmo Jamiolkowski, juntamente com Bialonczyk. e Zyczkowski [3], aprofundam a relação entre o teorema de Shmesh e a teoria da informação quântica, trazendo algumas aplicações envolvendo o espectro perimetral de canais quânticos com dois operadores de Kraus.…”