Upper bounds for the quantum fisher information?

Background:

The quantum fisher information can be used to lower bound the variance, when estimating a parameter $Theta$.
This estimation can be decomposed into three steps.

1. The initial state $rho_0$
2. The encoding process, discribed by a CPTP map $M_Theta$
3. The Messurement ${E_m}$ discribed by a POVM

The QFI allready contains an optimization about all possible POVMs.

My questions:

1. Is there an upperbound for the optimization over all possible
   initial states $max_{rho_0} QFI_{M_{Theta}}(rho_0)$ ?

2. Is there an upperbound for the optimization over all seprable
   initial states $max_{rho_0in text{SEP}}
   QFI_{M_{Theta}}(rho_0)$ ?

3. Are there other special cases, where bounds are known?