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### What are non-standard ways to describe the distance between states?

One nonstandard approach to measuring distances between states is the "$\varepsilon$-smooth relative complexity distance" $\mathcal C_\varepsilon(|\psi\rangle,|\phi\rangle)$ corresponding to ...
• 13.6k
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### How to derive the quantum Fisher information from the relative entropy?

Expressing the derivative $\partial_i\rho$ in terms of its eigenvalues and eigenvectors will show us that these two are not equal. I will assume a full-rank density matrix $\rho$ to streamline the ...
• 4,569
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### What is the conditional min-entropy for diagonal ("classical") matrices?

Long story short: taking $\sigma_B = \rho_B$ is equivalent to taking the worst case min-entropy $$\hat{H}_{\min}(A|B) = - \log \max_{a,b} P(A=a|B=b)\,,$$ and optimizing over $\sigma_B$ is equivalent ...
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• 6,226
1 vote
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• 26.3k
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### In what sense is the "conditional min-entropy" a conditional entropy?

Here is a perspective on why $H_{min}(A|B)$ is a min entropy, which may not directly answer your question. From an operational perspective, $H_{min}(A|B)$ is defined analogously to $H_{min}(A)$, at ...
• 7,448
1 vote

### What is the conditional min-entropy for diagonal ("classical") matrices?

Classical definition of $\mathsf D_{\rm max}(P\|Q)$ $\newcommand{\H}{\mathsf{H}}\newcommand{\Hmin}{\H_{\rm min}}\newcommand{\D}{\mathsf{D}}\newcommand{\Dmax}{\D_{\rm max}}$Consider the max-relative ...
• 26.3k
1 vote

### Data processing inequality for relative entropy in the presence of an amplitude damping channel

An easy (but perhaps cheating) answer: The relative entropy $S(A||B)$diverges when $A$ has support over the kernel of $B$ (e.g., wikipedia). Now, the kernel of \$B=|0\rangle^{\otimes n}\langle 0|^{\...
• 4,569

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