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If a have an initial (two-qubit) system in the state

$\rho_i= \begin{pmatrix} 0&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix}$

and this state evolves according to the evolution, $U=e^{-iHt}$ with

$H=A(\sigma_x\otimes\sigma_x+\sigma_y\otimes\sigma_y+\sigma_z\otimes\sigma_z)$,

I can find the evolution of qubit $1$ after some time, $t$, by taking the partial trace, which, for some matrix

$\rho_{AB}= \begin{pmatrix} a&b&c&d\\ e&f&g&h\\ i&j&k&l\\ m&n&o&p \end{pmatrix}$

is given by

$\rho_A= \begin{pmatrix} a+f&c+h\\ i+n&k+p \end{pmatrix}$.

For $\rho_i$, plotting the first element of this gives the evolution of qubit $1$ (shown in the Figure).enter image description here

I want to add a decoherence term to this first qubit only. I know the decay is exponential and happens on a $\mu$s timescale ($T_1=1\mu$s) and that the decay is such that the system approaches $0.5$ on the $y$ axis, however, I can't work out where to introduce this decay to give the result I am expecting. Any help would be appreciated!

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