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Non-lockability of quantum max-entropy

Let $D_{\alpha}(\rho\|\sigma):= \frac{1}{\alpha - 1} \log \mathrm{Tr}[\rho^\alpha \sigma^{1-\alpha}]$ be the Petz-Rényi divergence for $\alpha \in (0,1)\cup(1,\infty)$. Note that for $\alpha \in (0,1)\...
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Semi-definite program for conditional smooth max-entropy

Yes, you can formulate the smooth max-entropy as an SDP. The author of the book you linked notes this when they explain how to derive the SDP for the smooth min-entropy that you reference on page 91. ...
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3 votes
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Relating quantum max-relative entropy to classical maximum entropy

As far as I'm aware there isn't much of a meaningful connection. The corresponding entropy for $D_{\max}$ is the min-entropy (written $H_{\min}$ or $H_{\infty}$). It measures a sort of `worst case' ...
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3 votes
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Quasi concavity of max-relative entropy?

No, this is not possible. Consider $\rho_1 = \sigma_2 = \vert 0\rangle\langle 0 \vert$ and $\rho_2 = \sigma_1 = \vert 1\rangle\langle 1 \vert$. Then, $$D_{\max}(\rho_i\|\sigma_i) = \infty\quad \text{...
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  • 518
3 votes
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Continuity of Renyi entropies - limiting cases

Assuming everything is finite dimensional. For $S_0$ we have $$S_0(\rho) = \log \mathrm{rank}(\rho).$$ It's pretty straightforward to see this is not continuous. Take $\rho_{\epsilon} = \epsilon |0\...
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2 votes
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When can the max relative entropy be written as $D_{\max}(\rho\|\sigma) = \|\sigma^{-1/2}\rho\sigma^{-1/2}\|_{\infty}$?

There is a problem in the derivation you presented, since $\rho \leq \lambda \sigma$ is only equivalent to $\sigma^{-1/2} \rho \sigma^{-1/2} \leq \lambda I$ when $\sigma$ is invertible (or at least ...
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  • 106
2 votes
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Continuity bounds on $D_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B)$

Unfortunately $D_{\max}$ is not a continuous function and so functions built from it tend not to be continuous. For example consider consider the two states $$ \rho_{AB} = |00 \rangle \langle 00|, $$ ...
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2 votes
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Questions about the relation between max-relative entropy $D_{\max}(\rho||\sigma)$ and max-information

Can someone provide an example of a state $\rho_{AB}$ for which $\sigma^\star_B \neq \rho_B$? Why not start very easily, with a separable state such as $$ \rho_{AB}=\left(p_0|0\rangle\langle 0|\...
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1 vote
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Do we know the limits of the quantum Tsallis entropy?

Well for $q \to 0$ we have $$ \lim_{q \to 0} T_q(\rho) = \mathrm{rank}(\rho) - 1. $$ For $q \to \infty$ it's not really interesting as $$ \lim_{q \to \infty} T_{q}(\rho) = 0. $$ For the second result ...
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1 vote
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What is the relationship between these two definitions for the max-entropy?

The term max-entropy in quantum information is reserved for the following definition No it's not, many papers like https://arxiv.org/abs/0803.2770 use the term to refer to the quantity $\log \mathrm{...
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