I feel like I often see passing comments about this without proof in quantum info literature, but I wasn't able to show it myself! For instance p17/p18 of this classic
Lets say that we have some function $f : L_1 (H) \rightarrow L_1(H)$ on the space $L_1(H)$ of trace 1 density matrices on a finite dimensional complex Hilbert space. Lets say that I know that for any $p \in [0,1]$ then $f(p\rho + (1-p)\sigma) = p f(\rho) + (1-p)f(\sigma)$:
Does it follow that $f(a \rho + b \sigma) = af(\rho) + bf(\sigma)$ for any complex $a,b$?
If not, what if $f$ also preserves pure states, meaning there exists some function $f':H \rightarrow H$ such that for each $\psi$ then $f(|\psi\rangle \langle \psi|) = |f'(\psi)\rangle \langle f'(\psi)|$?