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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
  • 5,978
5 votes
Accepted

Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$

This only holds if the two distributions are independent. In this case $$ \begin{aligned} H_{\beta}(p \times q) &= \frac{1}{1-\beta} \log\left( \sum_{i,j}(p(i) q(j))^{\beta} \right) \\ &= \...
Rammus's user avatar
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4 votes

Why are all Rényi entropies equal for Clifford dynamics?

In stabilizer states (which are the states produced by clifford dynamics) the entanglement spectrum is flat, i.e., all the non-zero Schmidt coefficients are equal. Starting from there, it is ...
Norbert Schuch's user avatar
4 votes
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Why are all Rényi entropies equal for Clifford dynamics?

Let me report here a brief demonstration that I used to convince myself. The R'enyi entropies are \begin{equation*} S_\mathrm{A}^\alpha = \frac{1}{1-\alpha} \log_2 \text{Tr} (\rho_\mathrm{A}^{\...
gcemin's user avatar
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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\...
Rammus's user avatar
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3 votes

Clarification about inverses in sandwiched Renyi divergence

Firstly, the sandwiched divergence can be infinite even when $\rho$ and $\sigma$ are not orthogonal. For example, consider $\rho = \frac{|0\rangle \langle 0| + |1\rangle\langle 1|}{2}$ and $\sigma = |...
Rammus's user avatar
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2 votes

Clarification about inverses in sandwiched Renyi divergence

This quantity was defined independently in https://arxiv.org/abs/1306.1586, where many of its properties were explored therein. To answer your question, the quantity can be defined in the general case ...
Mark M. Wilde's user avatar
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, ...
Rammus's user avatar
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