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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?

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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.

The representation you suggest is the operator sum or Kraus representation.

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  • $\begingroup$ 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. $\endgroup$
    – CFRedDemon
    Commented May 6, 2019 at 11:17
  • $\begingroup$ $M_i$ are in general not unitary $\endgroup$
    – Agnieszka
    Commented Sep 19, 2020 at 17:17

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