Quantification and manipulation of magic states

Magic states can be used as a resource to circumvent the restrictions due to stabilizer-preserving operations, and magic-state conversion has not been studied in the single-copy regime thus far. Here we solve the question of whether a stabilizer-preserving quantum operation exists that can convert between two given magic states in the single-shot regime. We first phrase this question as a feasibility problem for a semi-definite program (SDP), which provides a procedure for constructing a stabilizer-preserving quantum operation (free channel) if it exists. Then we employ a variant of the Farkas Lemma to derive necessary and sufficient conditions for existence, and this method is used to construct a complete set of magic monotones.