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### What can be said about the non-negativity of the relative entropy of $S(\rho_{AB}||\rho_{B})$?

Source of the problem The purported contradiction arises due to the use of incorrect assumptions for Klein equality $$S(\rho||\sigma) \ge 0.$$ The inequality does not require any particular ...
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### Showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for classical-quantum states

As you say, $$\mathrm{Tr}[\rho_{XB} \log \rho_{XB}] = -S(X) + \sum_{x} p(x) \mathrm{Tr}[\rho_{B}^x \log \rho_B^x].$$ But if you can prove the above statement, then the exact same derivation gives ...

### How to take the limits of the sandwiched Renyi divergences?

Both limits are dealt with in a fair amount of detail in the work that originally defined the sandwiched entropies: On quantum Renyi entropies: a new generalization and some properties. In particular, ...
Long story short: taking $\sigma_B = \rho_B$ is equivalent to taking the worst case min-entropy $$\hat{H}_{\min}(A|B) = - \log \max_{a,b} P(A=a|B=b)\,,$$ and optimizing over $\sigma_B$ is equivalent ...
An easy (but perhaps cheating) answer: The relative entropy $S(A||B)$diverges when $A$ has support over the kernel of $B$ (e.g., wikipedia). Now, the kernel of \$B=|0\rangle^{\otimes n}\langle 0|^{\...