dorsal/arxiv
View SchemaThe Generalized Lyapunov Theorem and its Application to Quantum Channels
| Authors | Daniel Burgarth, Vittorio Giovannetti |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0605197 |
| URL | https://arxiv.org/abs/quant-ph/0605197 |
| DOI | 10.1088/1367-2630/9/5/150 |
| Journal | New J. Phys. 9 150 (2007) |
Abstract
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 exist 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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"abstract": "We give a simple and physically intuitive necessary and sufficient condition\nfor a map acting on a compact metric space to be mixing (i.e. infinitely many\napplications of the map transfer any input into a fixed convergency point).\nThis is a generalization of the \"Lyapunov direct method\". First we prove this\ntheorem in topological spaces and for arbitrary continuous maps. Finally we\napply our theorem to maps which are relevant in Open Quantum Systems and\nQuantum Information, namely Quantum Channels. In this context we also discuss\nthe relations between mixing and ergodicity (i.e. the property that there exist\nonly a single input state which is left invariant by a single application of\nthe map) showing that the two are equivalent when the invariant point of the\nergodic map is pure.",
"arxiv_id": "quant-ph/0605197",
"authors": [
"Daniel Burgarth",
"Vittorio Giovannetti"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1088/1367-2630/9/5/150",
"journal_ref": "New J. Phys. 9 150 (2007)",
"title": "The Generalized Lyapunov Theorem and its Application to Quantum Channels",
"url": "https://arxiv.org/abs/quant-ph/0605197"
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