Deterministic transformations of multilevel coherent states under incoherence-preserving operations

Quantum coherence, emerging from the 'superposition' of quantum states, is widely used in various information processing tasks. Recently, the resource theory of multilevel quantum coherence is attracting substantial attention. In this paper, we mainly study the deterministic transformations of resource pure states via free operations in the theoretical framework for multilevel coherence. We prove that any two multilevel coherent resource pure states can be interconverted with a nonzero probability via a completely positive and trace non-increasing $k$-coherence-preserving map. Meanwhile, we present the condition of the interconversions of two multilevel coherent resource pure states under $k$-coherence-preserving operations. In addition, we obtain that in the resource-theoretic framework of multilevel coherence, no resource state is isolated, that is, given a multilevel coherent pure state $|\psi\rangle$, there exists another multilevel coherent pure state $|\phi\rangle$ and a $k$-coherence-preserving operation $\Lambda_k$, such that $\Lambda_k(|\phi\rangle)=|\psi\rangle$.