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5 votes

What are explicit examples of smoothed conditional min(max) entropies?

I'll just give a classical example, which is a typical motivating example for these smooth quantities. Consider an $n$-bit distribution of the form $$ p(x) = \begin{cases} 1-\delta \qquad \text{ if }x=...
Rammus's user avatar
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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)\...
Rammus's user avatar
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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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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' ...
Rammus's user avatar
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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{...
rnva's user avatar
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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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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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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 ...
user13507's user avatar
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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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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|\...
DaftWullie's user avatar
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Which quantum entropies are meaningful with respect to continuous distributions of states?

I've found a partial answer for the case of conditional min-entropy, due to Ref. [1] (Appendix IV.B): Consider a fixed ensemble $\{(\rho_B(x), p(x))\}_{x \in \Sigma}$, where $p(x)$ is a probability ...
forky40's user avatar
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Difference between min/max-entropies and the von Neumann entropy

I'm not sure what you mean exactly with "small gap", but you can easily build examples where $H(A)$ and $H_{\rm max}(A)$ are "maximally different". For example, $$\rho = \begin{...
glS's user avatar
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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{...
user13507's user avatar
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