Questions tagged [renyi-entropies]
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Coherent information is a lower bound of channel capacity. What about coherent information based on Renyi entropies?
It is known that coherent information defined in terms of von Neumann entropies is a lower bound of quantum channel capacity. If we define coherent information in terms of $\alpha$-Renyi entropies, ...
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Why are all Rényi entropies equal for Clifford dynamics?
In this paper, by Adam Nahum et al., the authors trivially states that "For Clifford dynamics all Rényi entropies are equal ... " which is not trivial to me.
Is there a paper or lecture ...
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Clarification about inverses in sandwiched Renyi divergence
The sandwiched Renyi divergence is defined as in
$$
\tilde{D}_\alpha(\rho\|\sigma):=\frac{1}{\alpha−1}\log tr[(\sigma^{\frac{1−\alpha}{2\alpha}}\rho \sigma^{\frac{1−\alpha}{2 \alpha
}})^\alpha]
$$
The ...
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Find the minimal and maximal of $\widehat{S}_f (\rho \| U^* \sigma U)$
I have been study the minimal (maximal) of a $f-$divergence. Fumio Hiai introduced the $\widehat{S}_f (\rho \| \sigma)$ divergence in his article.
$$\widehat{S}_f (\rho \| \sigma) := \text{Tr} \sigma^{...
- 111
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What are explicit examples of smoothed conditional min(max) entropies?
Some general discussion of smoothed entropic quantities is found for example in Watrous notes, and an overview and discussion on its operational interpretations in (Koenig et al. 2008). It seems the ...
glS♦
- 23.4k
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Why are "smooth entropic quantities" useful/necessary?
Consider the $\epsilon$-smoothed relative max-entropy of $\rho$ with respect to $Q$, defined as (following Watrous' notation from these notes):
$$\mathrm D_{\rm max}^{\epsilon}(\rho\|Q) = \min_{\xi\in ...
glS♦
- 23.4k
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What is the idea behind compressibility results in terms of Renyi entropies?
In (Tomamichel 2015), in (1.2) the author mentions the result that a source $X$ with probability distribution $\rho\equiv\rho_X$ admits an $(\varepsilon,m)$-code as long as there is some $\alpha\in[\...
glS♦
- 23.4k
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How to take Statevector for subsystem?
I want to calculate the 2nd Renyi entropy using the density matrix in Qiskit. To do this, I need to calculate the $Tr(\rho^2)$ for subsystem. The complete system consists of 12 qubits from which I ...
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Understanding conditional $L_2$ distances
I see that conditional $L_2$ distances from uniform are defined in the following way:
$L_2(\rho_{AB}\vert \sigma_B)= \text{tr}\left(((\rho_{AB}- \mu_{A} \otimes \rho_{B}) (\mathbb{I}_A \otimes \...
- 469
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How to take the limits of the sandwiched Renyi divergences?
The sandwiched Renyi divergence is defined as
$$\begin{equation}
\tilde{D}_{\alpha}(\rho \| \sigma):=\frac{1}{\alpha-1} \log \operatorname{tr}\left[\left(\sigma^{\frac{1-\alpha}{2 \alpha}} \rho \...
- 11
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Continuity of Renyi entropies - limiting cases
The Renyi entropies are defined as
$$S_{\alpha}(\rho)=\frac{1}{1-\alpha} \log \operatorname{Tr}\left(\rho^{\alpha}\right), \alpha \in(0,1) \cup(1, \infty)$$
It is claimed that this quantity is ...
- 2,597
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Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$
The Renyi entropy of order $\beta$, for a discrete probability distribution $p$ is given by
\begin{equation}
H_{\beta}(p) = \frac{1}{1 - \beta} ~\log \left( \sum_{i \in S} p(i)^{\beta} \right),
\end{...
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