TY - JOUR

T1 - The generalized Lyapunov theorem and its application to quantum channels

AU - Burgarth, Daniel

AU - Giovannetti, Vittorio

PY - 2007/5

Y1 - 2007/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=35548960846&partnerID=8YFLogxK

U2 - 10.1088/1367-2630/9/5/150

DO - 10.1088/1367-2630/9/5/150

M3 - Article

SN - 1367-2630

VL - 9

SP - 1

EP - 18

JO - New Journal of Physics

JF - New Journal of Physics

M1 - 150

ER -