Maps Preserving Schatten p -Norms of Convex Combinations

Abstract

We study maps $\varphi$ of positive operators of the Schatten $p$-classes (), which preserve the $p$-norms of convex combinations, that is, ${\parallel t\rho +(\mathrm{1}-t)\sigma \parallel }_p={\parallel t\varphi (\rho )+(\mathrm{1}-t)\varphi (\sigma )\parallel }_p,\forall \rho ,\sigma \in {𝒮}_p^{+}{(H)}_1,t\in [\mathrm{0,1}]$. They are exactly those carrying the form $\varphi (\rho )=U\rho U^{\mathrm{*}}$ for a unitary or antiunitary $U$. In the case $p=\mathrm{2}$, we have the same conclusion whenever it just holds ${\parallel \rho +\sigma \parallel }_2={\parallel \varphi (\rho )+\varphi (\sigma )\parallel }_2$ for all the positive Hilbert-Schmidt class operators $\rho ,\sigma$ of norm $\mathrm{1}$. Some examples are demonstrated.