Channel-based coherence of quantum states

Quantum coherence is one of the most fundamental and striking features in quantum physics. Considered the standard coherence (SC), the partial coherence (PC) and the block coherence (BC) as variance of quantum states under some quantum channels (QCs) $\mathrm{\Phi }$, we propose the concept of channel-based coherence of quantum states, called $\mathrm{\Phi }$-coherence for short, which contains the SC, PC and BC, but does not contain the positive operator-valued measure (POVM)-based coherence. By our definition, a state $\rho$ is said to be $\mathrm{\Phi }$-incoherent if it is a fixed point of a QC $\mathrm{\Phi }$, otherwise, it is said to be $\mathrm{\Phi }$-coherent. First, we find the set ${ℐ}(\mathrm{\Phi })$ of all $\mathrm{\Phi }$-incoherent states for some given channels $\mathrm{\Phi }$ and prove that the set ${ℐ}(\mathrm{\Phi })$ forms a nonempty compact convex set for any channel $\mathrm{\Phi }$. Second, we define $\mathrm{\Phi }$-incoherent operations ($\mathrm{\Phi }$-IOs) and prove that the set of all $\mathrm{\Phi }$-IOs is a nonempty convex set. We also establish some characterizations of a $\mathrm{\Phi }$-IO in terms of its Kraus operators. Lastly, we discuss the problem of quantifying $\mathrm{\Phi }$-coherence and prove some related properties.