Transformations on Density Operators That Leave the Holevo Bound Invariant

For a given probability distribution λ ₁,…,λ _m we determine the structure of all such maps defined on a dense subset of density operators which leave the Holevo bound invariant i.e. which satisfy $$S\Biggl(\sum_{k=1}^m \lambda_k \phi(\rho_k)\Biggr)- \sum_{k=1}^m \lambda_k S\bigl(\phi (\rho_k)\bigr)= S\Biggl(\sum _{k=1}^m \lambda_k \rho_k\Biggr)- \sum_{k=1}^m \lambda_k S(\rho_k) $$ for all possible collections ρ ₁,…,ρ _m of density operators.