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The density matrix for an entangled state when both the qubits decohere with probability D1 and D2 in amplitude damping channel is given as $\rho_d$ in this paper on page 2. I am reproducing it here, $$ \begin{bmatrix} |\alpha^2|+|\beta^2|D_1 D_2 & 0 & 0 & \sqrt{\bar{D_1}\bar{D_2}}\alpha^*\beta \\ 0 & D_1 \bar{D_2}|\beta^2| & 0 & 0 \\ 0 & 0 & D_2 \bar{D_1}|\beta^2| & 0 \\ \sqrt{\bar{D_1}\bar{D_2}}\alpha\beta^* & 0 & 0 & \bar{D_2} \bar{D_1}|\beta^2| \end{bmatrix} $$ I am deriving the same expression using amplitude damping modeling, but I am not getting these non-diagonal terms $\sqrt{\bar{D_1}\bar{D_2}}\alpha \beta $ at matrix element $a_{34}$ and $a_{43}$ positions. Solving this using amplitude damping modeling we get, $\alpha \vert 0_S\rangle \vert 0_S\rangle \otimes \vert 0_E\rangle$ = $\alpha \vert 0_S\rangle\vert 0_S\rangle \otimes \vert 0_E\rangle$ similarly, \begin{equation} \begin{split} \beta \vert 1_S\rangle\vert 1_S\rangle \otimes \vert 0_E\rangle & = \beta \bar{D_1}\bar{D_2}\vert 1_S\rangle \vert 1_S\rangle \otimes \vert 0_E\rangle\\ &= \beta \bar{D_1}D_2\vert 1_S\rangle\vert 0_S\rangle \otimes \vert 0_E\rangle\\ &=\beta \bar{D_2}D_1\vert 0_S\rangle\vert 1_S\rangle \otimes \vert 0_E\rangle\\ &=\beta D_1 D_2\vert 0_S\rangle\vert 0_S\rangle \otimes \vert 0_E\rangle \end{split} \end{equation} Considering the density matrix of the above terms we get all the diagonal terms of the matrix. But I am not getting these non-diagonal terms. Please explain.

Thank you.

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