# Investigate noise in measures for Quantum-Non-Markovianity

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:

$\rho(t)=\Phi_t\rho(0)$

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

$\rho(t)=\Phi_{t,t_0}\rho(t_0)$

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

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

$<\hat{\epsilon}\rho(t)>_{\epsilon}=0$

$<\hat{\epsilon}\rho(t_1)\hat{\epsilon}\rho(t_2)>_{\epsilon}=\Gamma\delta(t_1-t_2)$

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:

$g(t)=\lim\limits_{\epsilon\rightarrow0^+}\frac{||{\left(\Phi_{t,t+\epsilon}\otimes\mathbb{1}\right)|{\Psi}><\Psi|}||}{\epsilon}$

where

$|\Psi>=\frac{1}{\sqrt{d}}\sum\limits_{i}|i>\otimes|i>$

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:

$\tilde{|\Psi>}:=\frac{1}{\sqrt{F}}\sum\limits_{i}\tilde{|i>}\otimes\tilde{|i>}$

where

$\tilde{|i>}:=\int\mathcal{N}\left\{|i>,\sigma\right\}|i>\mathrm{d}i$

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 $\tilde{g(t)}$

$\tilde{g(t)}=\lim\limits_{\epsilon\rightarrow0^+}\frac{||{\left(\Phi_{t,t+\epsilon}\otimes\mathbb{1}\right)\tilde{|\Psi>}\tilde{<\Psi|}||}}{\epsilon}$

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

$||{A}||:=Tr{\sqrt{A^{\dagger}A}}$

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