How do I plot the decoherence of an open system from its density matrix? If I have a two qubit state interacting with an environment that will decohere it, how do I model the decoherence from the density matrix? For example, if I start with some state
$\Psi(0)=|0>_1|1>_2\otimes|\psi_{e}>$ 
and act some unitary
$$U=e^{-iHt}$$
onto the density matrix of this state where $$H=A(\sigma_x\otimes\sigma_x+\sigma_y\otimes\sigma_y+\sigma_z\otimes\sigma_z)$$
where $\sigma_i$ are the Pauli matrices I will get some output density matrix $\rho(t)$. I then want to trace out everything but qubit 1 and use this output state $\rho_1(t)$ to find the coherence time of that particular spin. Is there a way to plot the decoherence time directly from this output density matrix? 
 A: The most general equation that describes the decoherence of a density matrix is the Lindblad equation. It has the form
$$ \partial_t \rho = - [H_0, \rho] + \sum_n \gamma_n (L_n \rho L_n^{\dagger} - L_n^{\dagger} L_n \rho - \rho L_n^{\dagger} L_n) . $$
For the qubit case you can express the Lindblad operators $L_n$ in terms of Pauli matrices. So by playing around with different such operators and solving the equation, you may get a feeling of how it behaves and see what the effect is on the density matrix.
A: You can ignore the system $e$, because it does not interact with system $1$. Up to an irrelevant global phase of $e^{i A t}$ we have $$|\psi(t)\rangle = \cos(2 A t) |01\rangle - i \sin(2 A t) |1 0\rangle$$ and the reduced state is $$\rho_A=\left(
\begin{array}{cc}
 \cos ^2(2 A t) & 0 \\
 0 & \sin ^2(2 A t) \\
\end{array}
\right).$$
Now as user Count Iblis suggested, you can look at the purity of this state, which is $\mathrm{tr}(\rho_A^2)=(3+\cos(8 A t))/4$. Note how the purity oscillates. This shows that depending on the interaction time and strength the two systems are entangled or not.   
Unfortunately your state $|\psi(0)\rangle$ is incoherent, so I think there are better examples to illustrate the above. Consider instead the state $$|\psi(0)\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes |0\rangle,$$ which is maximally coherent with respect to the computational basis, and the Hamiltonian $$H=\frac{\pi}{4}(\sigma_0-\sigma_z)\otimes (\sigma_0-\sigma_x),$$
where $\sigma_0$ is the identity matrix. The reduced state is
$$\rho_A=\frac{1}{4}\left(
\begin{array}{cc}
 2 & 1+e^{-i \pi  t} \\
 1+e^{i \pi  t} & 2 \\
\end{array}
\right).$$
Here you see how the coherence, i.e. the absolute value of the off-diagonal elements, oscillates with time. In particular, for $t=1$ (where the interaction is the controlled-Not gate) the reduced state has no coherence at all.
