Quantifying dynamical total coherence in a resource non-increasing framework

We quantify the dynamical quantum resource in the resource non-increasing (RNI) framework, namely the free dynamical resource is defined by the channels that cannot increase the static "resourcefulness" of any input state. We present two kinds of approaches to quantifying the dynamical resource, the distance measures and the maximal increasing static resource. As a demonstration, we quantify the dynamical total coherence with our presented measures. It is shown that the distance-based measures have good operational interpretation through quantum processing tasks and can be numerically calculated by semidefinite programming (SDP) and the measures based on the maximal increasing static resource could lead to the analytical solution. As an application, we consider the dynamical total coherence of the qubit amplitude damping channel. Both the analytical measure based on the static l(2) norm and the numerical illustrations based on the SDP are given.