# Tag Info

5

The idea is that $$\frac{d}{dt}\rho=\mathcal{L}\rho\qquad \Leftrightarrow\qquad \rho(t)=\exp(t\mathcal{L})\rho(0).$$ In that sense, the Lindbladian $\mathcal{L}$ generates evolution through $$\rho(dt)\approx \rho(0)+dt \mathcal{L}\rho(0),$$ which is the same sense as an infinitesmal generator of a Lie group. You might be familiar with something like a ...

4

The given form of the Kraus operators, though not unique, tells us what is physically happening. In the case of pure-state outputs, each Kraus operator takes one of the possible basis states and swaps it with the desired output state. All basis states thus become $|0\rangle$ through some sort of swapping mechanism. In the case of mixed-state outputs, each ...

4

I think this question is generally difficult because there is no standard metric for non-Markovianity, for example this paper would suggest you try to express the evolution in a time-local canonical (Lindblad) form and then look at the negativity of the rates, but other metrics may not agree for certain channels. Perhaps it is easier to answer the simpler ...

3

Go back to the Lindblad master equation: $$\frac{d\rho}{dt}=i[H,\rho]+\sum_nL_n^\dagger\rho L_n-\frac12\sum_nL_nL_n^\dagger \rho-\frac12\sum_n\rho L_nL_n^\dagger.$$ The statement that the maximally mixed state is a fixed point is equivalent to saying that if $\rho=I/d$ then $\frac{d\rho}{dt}=0$. So, that's check that. \begin{align*} d\frac{d\rho}{dt}&=...

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I'll show you how to do it by brute force, since this will demonstrate a lot of techniques that will be useful for you if you have to derive something more complicated. The Lindblad evolution:  \frac{\textrm{d}\rho}{\textrm{d}t} = -\frac{\textrm{i}}{\hbar}[H,\rho] + \sum_{\mu,\nu} h_{\mu,\nu} \left( L_\mu \rho L_\nu^\dagger -\frac{1}{2}\left\{ L^\dagger_\...

3

I guess from the operator sum representation $\rho(t) = \sum_k K_k \rho(t_0) K_k^\dagger$ alone you won't be able to make any statement about non-Markovianity, since you are missing interesting features: (Domain) Is this dynamical map valid for all initial reduced density matrices? If so, the initial state might be a product state $\rho(t_0) \otimes \rho_E$ ...

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