2006/05/31 by Daniel Burgarth, Vittorio Giovannetti · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum Information and Cryptography #advanced mathematical theories #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1088/1367-2630/9/5/150
published as New J. Phys. 9 150 (2007) · 13 pages, 3 figures
arxiv created 2007/03/09 · openalex publication_date 2007/05/23 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
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.