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arXiv:2307.13694 · Lobachevskii Journal of Mathematics

Compactness Criterion for Families of Quantum Operations in the Strong Convergence Topology and Its Applications

Максим Евгеньевич Широков

Published Jun 1, 2024• 0 citations
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Abstract

Abstract A revised version of the compactness criterion for families of quantum operations in the strong convergence topology obtained in [1] is presented, along with a more detailed proof and the examples showing the necessity of this revision. Several criteria for the existence of limit points of a sequence of quantum operations w.r.t. the strong convergence are obtained and discussed. Applications in different areas of quantum information theory are described.

DOI
10.1134/s1995080224603151

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Abstract A revised version of the compactness criterion for families of quantum operations in the strong convergence topology obtained in [1] is presented, along with a more detailed proof and the examples showing the necessity of this revision. Several criteria for the existence of limit points of a sequence of quantum operations w.r.t. the strong convergence are obtained and discussed. Applications in different areas of quantum information theory are described.
We give a simple and physically intuitive necessary and sufficient condition for a map acting on a compact metric space to be mixing (i.e. infinitely many applications of the map transfer any input into a fixed convergency point). This is a generalization of the 'Lyapunov direct method'. First we prove this theorem in topological spaces and for arbitrary continuous maps. Finally we apply our theorem to maps which are relevant in open quantum systems and quantum information, namely quantum channels. In this context, we also discuss the relations between mixing and ergodicity (i.e. the property that there exists only a single input state which is left invariant by a single application of the map) showing that the two are equivalent when the invariant point of the ergodic map is pure.
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