  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.
