# What is the quantum capacity of the combined amplitude and phase damping channel?

Quantum capacity for the amplitude damping channel and the pure dephasing channel have closed-form formulas as it can be seen in section 24.7.2 of From Classical to Quantum Shannon Theory. However, I am unsure if there exist such a result for the more complete decoherence model that combines both actions into the combined amplitude and phase damping channel. This is a Kraus rank $$3$$ quantum channels with the following error operators \label{eq:ampphaseKraus} \begin{aligned} E_0 &= \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma-(1-\gamma)\lambda} \end{pmatrix} \\ \\E_1 &= \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix} \\E_2 &= \begin{pmatrix} 0 & 0 \\ 0 & \sqrt{(1-\gamma)\lambda} \end{pmatrix} \end{aligned} where $$\gamma$$ is the damping parameter and $$\lambda$$ is the scattering parameter. This channel can be seen as the serial composition of the composing ampitude damping channel and dephasing channel as $$\mathcal{N}_{\mathrm{APD}}=\mathcal{N}_{\mathrm{PD}}\circ \mathcal{N}_{\mathrm{AD}}$$. I am wondering if there is some known result on the capacity of this channel, as it happens for its constituting channels.