dorsal/arxiv
View SchemaThere, and Back Again: Quantum Theory and Global Optimisation
| Authors | Koenraad M. R. Audenaert |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0402076 |
| URL | https://arxiv.org/abs/quant-ph/0402076 |
| Journal | Proceedings of Sixteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS2004), Catholic University of Leuven, Belgium, 5-9 July 2004. |
Abstract
We consider a problem in quantum theory that can be formulated as an optimisation problem and present a global optimisation algorithm for solving it, the foundation of which relies in turn on a theorem from quantum theory. To wit, we consider the maximal output purity $\nu_q$ of a quantum channel as measured by Schatten $q$-norms, for integer $q$. This quantity is of fundamental importance in the study of quantum channel capacities in quantum information theory. To calculate $\nu_q$ one has to solve a non-convex optimisation problem that typically exhibits local optima. We show that this particular problem can be approximated to arbitrary precision by an eigenvalue problem over a larger matrix space, thereby circumventing the problem of local optima. The mathematical proof behind this algorithm relies on the Quantum de Finetti theorem, which is a theorem used in the study of the foundations of quantum theory. We expect that the approach presented here can be generalised and will turn out to be applicable to a larger class of global optimisation problems. We also present some preliminary numerical results, showing that, at least for small problem sizes, the present approach is practically realisable.
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"abstract": "We consider a problem in quantum theory that can be formulated as an\noptimisation problem and present a global optimisation algorithm for solving\nit, the foundation of which relies in turn on a theorem from quantum theory. To\nwit, we consider the maximal output purity $\\nu_q$ of a quantum channel as\nmeasured by Schatten $q$-norms, for integer $q$. This quantity is of\nfundamental importance in the study of quantum channel capacities in quantum\ninformation theory. To calculate $\\nu_q$ one has to solve a non-convex\noptimisation problem that typically exhibits local optima. We show that this\nparticular problem can be approximated to arbitrary precision by an eigenvalue\nproblem over a larger matrix space, thereby circumventing the problem of local\noptima. The mathematical proof behind this algorithm relies on the Quantum de\nFinetti theorem, which is a theorem used in the study of the foundations of\nquantum theory.\n We expect that the approach presented here can be generalised and will turn\nout to be applicable to a larger class of global optimisation problems. We also\npresent some preliminary numerical results, showing that, at least for small\nproblem sizes, the present approach is practically realisable.",
"arxiv_id": "quant-ph/0402076",
"authors": [
"Koenraad M. R. Audenaert"
],
"categories": [
"quant-ph"
],
"journal_ref": "Proceedings of Sixteenth International Symposium on Mathematical\n Theory of Networks and Systems (MTNS2004), Catholic University of Leuven,\n Belgium, 5-9 July 2004.",
"title": "There, and Back Again: Quantum Theory and Global Optimisation",
"url": "https://arxiv.org/abs/quant-ph/0402076"
},
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