Say we have two quantum channels $\mathcal{E}_1$ and $\mathcal{E}_2$ with minimum fidelities $\mathcal{F}_1$ and $\mathcal{F}_2$. Can we place any bounds on the fidelity of the channel $\mathcal{E}=\mathcal{E}_1\circ \mathcal{E}_2$ (let's call the fidelity of this channel $\mathcal{F}$)? Is it possible to show something like $\mathcal{F}\geq \mathcal{F}_1\mathcal{F}_2$ for a general case?

Say we have two quantum channels $\mathcal{E}_1$ and $\mathcal{E}_2$ with channel fidelities $\mathcal{F}_1$ and $\mathcal{F}_2$. Can we place any bounds on the fidelity of the channel $\mathcal{E}=\mathcal{E}_1\circ \mathcal{E}_2$ (let's call the fidelity of this channel $\mathcal{F}$)? Is it possible to show something like $\mathcal{F}\geq \mathcal{F}_1\mathcal{F}_2$ for a general case?

I define channel fidelity as: $\mathcal{F}(\mathcal{E})=\textrm{min}_\rho~Tr(\rho,\mathcal{E}(\rho)),$ similar to the definition in Nielsen and Chuang.