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There, and Back Again: Quantum Theory and Global Optimisation

Koenraad M. R. Audenaert
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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 $ν_q$ 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 $ν_q$ 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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Koenraad M. R. Audenaert. 2004-02-16. There, and Back Again: Quantum Theory and Global Optimisation. https://arxiv.org/abs/quant-ph/0402076

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