What is the idea behind quantum speed limits? Could someone please explain to me how the very basic idea behind existence of a quantum speed limit arises?
I think I understand (if it's correct) how it arises naturally between two pure states since two pure states may be time evolved into one another via many different Schroedinger equations (created by different Hamiltonians). The time for the evolution varies with the different Hamiltonians and thus there may be a lower bound on the time required.
But how does this problem arise with mixed states?
 A: Quantum speed limits are not something particularly mysterious, albeit the topical literature can admittedly be confusing in this regard. I find the best way to think of them is as methods to easily lower bound, for a given dynamics, the time a given input state will take to reach a given output state.
Note that quantum speed limits, at least in their standard formulation, are a function of an input state $\psi_i$, an output state $\psi_f$, and the dynamics (be it an Hamiltonian $H$ or a more general dynamical map) leading one to the other. That information, as you correctly point out yourself, is sufficient to fully define the time $t$ required to go from $\psi_i$ to $\psi_f$.
So, if given $\psi_i,\psi_f,H$ we can find $t$ (or more precisely the smallest such $t$) such that $\psi_i\to\psi_f$, what's the point of quantum speed limits? Well, as per my initial paragraph, the point is simply that they give you an easy-to-compute lower bound for this time. It's a way to easily figure out going from $\psi_i$ to $\psi_f$ will require at least a certain amount of time, without actually having to solve the (possibly quite complicated) dynamics.
