I'm currently working on investigating noise and how it effects measures for non-markovianity in quantum systems...the problem is that i'm not really used to 'doing science' on my own (just recently started my masterthesis).

I have a few proposals of what i would like to try, but need some help in how far these concepts would make sense. I would be glad if you could tell me what to discard and where to investigate further.

As an overview:

I have the differential equation of my density operator (of my system):

$\frac{\partial\rho(t)}{\partial t}=\mathcal{L}_t\rho(t)$

For now let's assume that the map is CPT (completely positive and trace preserving) and that the dynamics is divisible (or even CP-divisible), defined by:


$\Phi_{t,t_0}=\Phi_t\Phi_{t_0}^{-1}$, $t>t_0$


One thing i would like to try is to add some gaussian noise on the dynamical level, aka on the differential equation for $\rho(t)$:

$\frac{\partial\rho(t)}{\partial t}=\left(\mathcal{L}_t+\hat{\epsilon}\right)\rho(t)$



Is this even a ansatz that makes sense? What would be the next step? How could i investigate what happens with the CP-divisibility of the original process? what would be interesting to do?

Another thing I'd like to try is to investigate in this measure for non-markovianity:




is the maxially entangles state between the system and an ancilla. $g(t)$ is zero if the map $\Phi_{t}$ is CP-divisible, or in other words if $\Phi_{t,t+\epsilon}$ is completely positive.

I'd like to implement the following state:




is a state centered at $|i>$ and normally distributed around it with some standard deviation.

I then want to investigate what happens with the new $\tilde{g(t)}$


Another problem I have is how the norm is defined on the tensor-product space:


$||A\otimes B||=Tr{\sqrt{\left(A \otimes B\right)^{\dagger}\left(A \otimes B\right)}}=Tr{\sqrt{\left(A^{\dagger}A\right)\otimes\left(B^{\dagger}B\right)}}=?$

Thanks in regard already!


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