Exact solution for the max-min quantum error recovery problem

This paper considers the max-min quantum error recovery problem; the recovery channel to be designed maximizes the fidelity between input and output states of a given noisy channel, while the minimum is taken over all possible pure input states. In general, this kind of max-min problem is cast as a non-convex optimization problem and is thus very hard to solve even with the aid of high-quality computational tools. Nevertheless, it is shown that, when the input takes a qubit, the problem is exactly convex for any size of error process. The Sum of Squares (SOS) characterization of a specific class of polynomial functions plays a crucial role in deriving this result.