3
votes
Accepted
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 ...
- 14.4k
2
votes
Accepted
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 ...
- 4,202
2
votes
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, ...
- 4,202
2
votes
Accepted
What is the conditional min-entropy for diagonal ("classical") matrices?
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 ...
- 4,202
1
vote
Data processing inequality for relative entropy in the presence of an amplitude damping channel
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|^{\...
- 2,159
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