# Superoperator cannot increase relative entropy

So I have to show that a superoperator $$\$$ cannot increase relative entropy using the monotonicity of relative entropy:

$$S(\rho_A || \sigma_A) \leq S(\rho_{AB} || \sigma_{AB}).$$

What I have to prove:

$$S(\\rho|| \ \sigma) \leq S(\rho || \sigma).$$

Now the hint is that I should use the unitary representation of the superoperator $$\$$. I know that we can represent $$\ \rho = \sum_i M_i \rho M_i^{\dagger}$$ with $$\sum_i M_i M_i^{\dagger} = I$$. Now I am able to write out $$S(\\rho|| \ \sigma_A)$$ in this notation, but that doesn't bring me any further.

Does anyone have any idea how to show this in the way that the questions hints to? I already read the original paper of Lindblad but this doesn't help me (he does it another special way). Any clues or how to do this?

Every superoperator acting on a system $$A$$ can be represented by the following operation: first add some ancilla system $$B$$, then do a unitary on the combined system $$AB$$. Finally, trace out the system $$B$$. This is the Unitary or Stinespring representation. This allows you to sequentially use your first equation, and your desired result should follow.
• Thank you so much. I was confused because the $M_i$ were also unitary theirselves so I thought I had to use the Kraus rep. Found it now. Commented May 6, 2019 at 11:17
• $M_i$ are in general not unitary Commented Sep 19, 2020 at 17:17