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 $\mathcal{A}(A_1,\dots A_K)$. By applying the Shemesh and Amitsur - Levitzki theorems to analyse the structure of the algebra $\mathcal{A}(A_1,\dots A_K)$ one can predict the asymptotic dynamics for a class of operations.

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Cited by in corpus (3)

• Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states
• Statistics of projective measurement on a quantum probe as a witness of noncommutativity of algebra of a probed system
• Peripherally automorphic unital completely positive maps