# Questions tagged [min-entropy]

The smallest of the Rényi family of entropies, corresponding to the most conservative way of measuring the unpredictability of a set of outcomes, as the negative logarithm of the probability of the most likely outcome.

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### Difference between min/max-entropies and the von Neumann entropy

Consider the (smooth) min-entropy, max-entropy and von Neumann entropy of a given density operator $\rho_A$. Does a small gap between $H_{\max(\min)}(A)_\rho$ and $H(A)_\rho$ implies a small gap ...
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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 ...
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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 ...
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### Data Processing equality variation

Let $\rho_{AB}$ be a state and $T: B \rightarrow C$ be a CPTP map with $\sigma_{AC}= T(\rho_{AB})$. It is well known that $H_{\infty}(A \vert B)_{\rho} \geq H_{\infty}(A \vert C)_{\sigma}$ (aka data ...
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### How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?

I create a quantum state $| \psi \rangle = \frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ for a randomly chosen $x_0,x_1$ of 50 bits. I give this quantum state $|\psi \rangle$ to you and you return ...
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### Prove that for a cq-state $\rho_{XE}$, $H_\infty(X|E) \ge H_\infty(X) - \log|E|$

Given a classical-quantum(cq) state $\rho_{XE}$, where the $X$ register is classical, I want to prove the following: \begin{align} H_\infty(X|E) \ge H_\infty(X) - \log|E|,\tag{1} \end{align} i.e....
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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 ...
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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 ...
Suppose we have a classical quantum state $\sum_x |x\rangle \langle x|\otimes \rho_x$, one can define the smooth-min entropy $H_\min(A|B)_\rho$ as the best probability of guessing outcome $x$ given \$\...