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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 \begin{equation}\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} \end{equation} 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.

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