Optimal decomposition of incoherent qubit channel

Abstract: We show that any incoherent qubit channel could be decomposed into four incoherent Kraus operators. The proof consists in showing existence of four incoherent Kraus operators by decomposing the corresponding Choi-Jamio\l{}kowski-Sudarshan matrix. We mention some applications of this optimal decomposition. We also show that the Kraus rank and incoherent rank are different even for qubit channel.