  Recently, a lot of attention has been devoted to finding physically
realisable operations that realise as closely as possible certain desired
transformations between quantum states, e.g. quantum cloning, teleportation,
quantum gates, etc. Mathematically, this problem boils down to finding a
completely positive trace-preserving (CPTP) linear map that maximizes the
(mean) fidelity between the map itself and the desired transformation. In this
note we want to draw attention to the fact that this problem belongs to the
class of so-called semidefinite programming (SDP) problems. As SDP problems are
convex, it immediately follows that they do not suffer from local optima.
Furthermore, this implies that the numerical optimization of the CPTP map can,
and should, be done using methods from the well-established SDP field, as these
methods exploit convexity and are guaranteed to converge to the real solution.
Finally, we show how the duality inherent to convex and SDP problems can be
exploited to prove analytically the optimality of a proposed solution. We give
an example of how to apply this proof method by proving the optimality of Hardy
and Song's proposed solution for the universal qubit $\theta$-shifter
(quant-ph/0102100).
