The Fidelity of Recovery is Multiplicative

Fawzi and Renner (arXiv:1410.0664) recently established a lower bound on the conditional quantum mutual information of tripartite quantum states $\rho_{ABC}$ in terms of the fidelity of recovery, i.e. the maximal fidelity of the state $\rho_{ABC}$ with a state reconstructed from its marginal $\rho_{BC}$ by acting only on the $C$ system. In this brief note we show that the fidelity of recovery is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandao et al. (arXiv:1411.4921) of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions.