The Transformation of Quantum States in Coherence Theory

We study the transformation between quantum states under incoherent operations and unitary incoherent operations. In high dimensional systems, we show any two pure states can be transformed into each other under incoherent operations if and only if they can be transformed into each other under unitary incoherent operations. For mixed states, we show any two incoherent states can be transformed into each other under incoherent operations. Furthermore, we show any incoherent states can be transformed by any mixed states under incoherent operations. In qubit systems, we prove the necessary and sufficient condition for the transformation between any two mixed states by incoherent operations alternatively and obtain the corresponding incoherent operations. Moreover we prove any two mixed qubit states with nonzero coherence can be transformed into each other under incoherent operations if and only if they can be transformed into each other under unitary incoherent operations.