Relation between optimal decompositions for entanglement of formation In the answer of this question, the last paragraph says that

If you know one decomposition which is optimal for Entanglement of Formation for a given state $\rho$, you can obtain the optimal decomposition for other states by simply shifting the weights $q_i$ in the optimal decomposition.

I'd like to know how to prove this or is there any references?
 A: We will prove the following statement:

Let $\rho=\sum p_i \rho_i$ ($p_i>0$) be a decomposition of a given state $\rho$ which minimizes the cost function $C(\{p_i,\rho_i\})=\sum p_i f(\rho_i)$ for that state. Then, 
  $\rho'= \sum p_i' \rho_i$ is a decomposition which minimizes $C$ for the state $\rho'$.

Note. In the case of entanglement of formation, $f(\rho_i)=S(\mathrm{tr}_B\,\rho_i)$.
Proof.  We will prove this by contradiction. Assume that there exists a decomposition $\rho'=\sum q_j \sigma_j$ for which 
$$\sum q_j\,f(\sigma_j)<\sum p_i'\,f(\rho_i)\ .$$
Choose a $\lambda>0$ s.th. $r_i:=p_i-\lambda p_i'\ge0$ for all $i$.  Then, 
\begin{align}
C(\{p_i,\rho_i\}) & = \sum_i p_i f(\rho_i)\\
&=\lambda\left[\sum_i p_i'\, f(\rho_i)\right] + \sum_i r_i\, f(\rho_i)\\
&>\lambda\left[\sum_j q_j\,f(\sigma_j)\right]+ \sum_i r_i\, f(\rho_i)\\
&=C(\{w_k,\tau_k\})
\end{align}
with the ensemble $\{w_k,\tau_k\}$ the union of the ensembles $\{\lambda q_j,\sigma_j\}$ and $\{r_i,\rho_i\}$, which is in contradiction to the assumption that $\rho=\sum p_i\rho_i$ is the optimal decomposition for $\rho$.
$\Box$
As far as I am aware, this is a well-known result, but I don't know where this is written down.

Late edit: I just stumbled across the result: It is proven in Sec. IV B of K.~G. H. Vollbrecht and R. F. Werner, Phys. Rev. A 64, 062307 (2001) (also quant-ph/0010095).
