Questions tagged [relative-entropy]
For questions about the (quantum) relative entropy, the quantum version of the classical Kullback-Leibler divergence.
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Does the quantum relative entropy have a direct operational interpretation?
Consider the quantum relative entropy, defined as
$$D(\rho\|\sigma) = \operatorname{tr}(\rho\log\rho)-\operatorname{tr}(\rho\log\sigma),$$
for all $\rho,\sigma\ge0$ such that $\operatorname{im}(\rho)\...
glS♦
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Conditional entropy as relative entropy between probability distributions
Find the expression for the conditional entropy $H(Y|X)$ as a relative entropy between two probability distributions. Use this expression to deduce that $H(Y |X)≥0$, and to find the equality ...
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How to derive the quantum Fisher information from the relative entropy?
The quantum relative entropy (QRE) between two states $\rho$ and $\sigma$ is given by
$$
S(\rho\|\sigma)=\operatorname{Tr}(\rho\ln\rho)-\operatorname{Tr}(\rho\ln\sigma)
$$
Now if $\rho$ and $\sigma$ ...
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What is the quantum relative entropy between pure states?
Given two pure quantum state $\rho=|\psi_\rho\rangle\langle\psi_\rho\mid$ and $\sigma=\mid\psi_\sigma\rangle\langle\psi_\sigma\mid$ ($\rho\neq\sigma$).
We know that the fidelity between quantum ...
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In what sense is the "conditional min-entropy" a conditional entropy?
$\newcommand{\H}{\mathsf{H}}\newcommand{\Hmin}{\H_{\rm min}}\newcommand{\D}{\mathsf{D}}\newcommand{\Dmax}{\D_{\rm max}}$Consider the conditional min-entropy $\Hmin(A|B)_\rho$, discussed e.g. in this ...
glS♦
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What is the conditional min-entropy for diagonal ("classical") matrices?
The conditional min-entropy, discussed e.g. in these notes by Watrous, as well as in this other post, can be defined as
$$\mathsf{H}_{\rm min }(\mathsf{X} \mid \mathsf{Y})_{\rho}\equiv -\inf _{\sigma \...
glS♦
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Data processing inequality for relative entropy in the presence of an amplitude damping channel
Consider the single qubit quantum depolarizing channel, given by
$$T(\rho) = (1- p)\rho + p \frac{\mathbb{I}}{2}. $$
For an $n$ qubit state $\rho$, according to Definition 6.1 of this paper, the ...
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Quasi concavity of max-relative entropy?
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$
It is known that the max-relative entropy is quasi-convex. ...
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Showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for classical-quantum states
Having some trouble showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for $\rho_{XB}=\sum_{x}p(x)|x\rangle\langle x|\otimes\rho_{B}^{x}$ and $\sigma_{XB}=\sum_{x}p(x)...
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How to take the limits of the sandwiched Renyi divergences?
The sandwiched Renyi divergence is defined as
$$\begin{equation}
\tilde{D}_{\alpha}(\rho \| \sigma):=\frac{1}{\alpha-1} \log \operatorname{tr}\left[\left(\sigma^{\frac{1-\alpha}{2 \alpha}} \rho \...
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What can be said about the non-negativity of the relative entropy of $S(\rho_{AB}||\rho_{B})$?
Taking $\rho_{AB}=\rho_{A}\otimes \rho_{B}$, where $S(\rho_{A})$ and $S(\rho_{B})$ aren't 0, it's easy to see that
$$S(\rho_{AB}||I \otimes \rho_{B})=-S(\rho_{A})-S(\rho_{B})+S(\rho_{B})=-S(\rho_{A}).$...
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When can the max relative entropy be written as $D_{\max}(\rho\|\sigma) = \|\sigma^{-1/2}\rho\sigma^{-1/2}\|_{\infty}$?
The max-relative entropy between two states is defined as $D_{\max}(\rho\|\sigma) = \log\lambda$, where $\lambda$ is the smallest real number that satisfies $\rho\leq \lambda\sigma$, where $A\leq B$ ...
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Is the quantum min-relative entropy $D_{\min}(\rho\|\sigma)=-\log(F(\rho, \sigma)^2)$ or $D_{\min}(\rho\|\sigma)=-\log(tr(\Pi_\rho\sigma))$?
In John Watrous' lectures, he defines the quantum min-relative entropy as
$$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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How to calculate the conditional min-entropy via a semidefinite program?
I am trying to formulate the calculation of conditional min-entropy as a semidefinite program. However, so far I have not been able to do so. Different sources formulate it differently. For example, ...
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