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# Questions tagged [relative-entropy]

For questions about the (quantum) relative entropy, the quantum version of the classical Kullback-Leibler divergence.

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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 $\sigma$ ...
• 93
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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 ...
• 51
165 views

### 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 ...
• 2,997
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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, ...
• 1,785
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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 ...
• 93
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• 831
1 vote
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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 ...
• 831
1 vote
160 views

### 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 ...
• 25.4k
1 vote
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### How to calculate relative entropy of coherence?

Relative entropy of coherence for a density matrix p is defined as follow $C(p)=S(p_{diag})-S(p)$ Where S is the von neumann entropy. for more info check the link (look at the section result) I know ...
• 733
1 vote