Optimal Binary Codes and Measurements for Classical Communication over Qubit Channels

We propose constructive approaches for the optimization of binary classical communication over a general noisy qubit quantum channel, for both the error probability and the classical capacity functionals. After showing that the optimal measurements are always associated to orthogonal projections, we construct a parametrization of the achievable transition probabilities via the coherence vector representation. We are then able to rewrite the problem in a form that can be solved by standard, efficient numerical algorithms and provides insights on the form of the solutions.