# Questions tagged [relative-entropy]

For questions about the (quantum) relative entropy, the quantum version of the classical Kullback-Leibler divergence.

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### Data processing inequality for relative entropy in the presence of an amplitude damping channel

Consider the single qubit quantum depolarizing channel, given by $$T(\rho) = (1- p)\rho + p \frac{\mathbb{I}}{2}.$$ For an $n$ qubit state $\rho$, according to Definition 6.1 of this paper, the ...
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### When can the max relative entropy be written as $D_{\max}(\rho\|\sigma) = \|\sigma^{-1/2}\rho\sigma^{-1/2}\|_{\infty}$?

The max-relative entropy between two states is defined as $D_{\max}(\rho\|\sigma) = \log\lambda$, where $\lambda$ is the smallest real number that satisfies $\rho\leq \lambda\sigma$, where $A\leq B$ ...
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### Is the quantum min-relative entropy $D_{\min}(\rho\|\sigma)=-\log(F(\rho, \sigma)^2)$ or $D_{\min}(\rho\|\sigma)=-\log(tr(\Pi_\rho\sigma))$?

In John Watrous' lectures, he defines the quantum min-relative entropy as $$D_{\min}(\rho\|\sigma) = -\log(F(\rho, \sigma)^2),$$ where $F(\rho,\sigma) = tr(\sqrt{\rho\sigma})$. Here, I use this ...
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### Questions about the relation between max-relative entropy $D_{\max}(\rho||\sigma)$ and max-information

The max-relative entropy between two states is defined as $$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$ where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
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