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.
Questions tagged [min-entropy]
18 questions
8
votes
1 answer
How does the conditional min-entropy $H_{\rm min}(A|B)_\rho$ relate to the conditional entropy $H(X|Y)_\rho$?
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 $\rho_x$. How does this quantity relate to…
john_smith
- 81
- 1
5
votes
1 answer
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, in this highly influential paper, it has been…
QuestionEverything
- 1,837
- 12
- 23
5
votes
1 answer
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 me back $x_0$. How do I prove that conditioned on…
nishkr
- 113
- 5
5
votes
1 answer
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 question and answer to make the definition simpler…
James
- 51
- 1
4
votes
1 answer
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 quantity was introduced in (Renner and Wolf 2004),…
glS
- 27,510
- 7
- 37
- 125
3
votes
2 answers
What is the conditional min-entropy of a pure bipartite state?
In this paper, it is stated that the conditional min-entropy $H(A|B)_{\rho_{AB}}$ of $A$ conditioned on $B$ for any $\textbf{pure}$ quantum system $\rho_{AB}=|\psi_{AB} \rangle \langle \psi_{AB} |$ is
$$ H_{\textrm{min}}(A|B)_{|\psi_{AB} \rangle…
quantum_theo
- 55
- 5
3
votes
1 answer
Which quantum entropies are meaningful with respect to continuous distributions of states?
When using a quantum channel to transmit classical information, we consider an ensemble $\mathcal{E} = \{(\rho_x, p(x))\}$ consisting of states $\rho_x$ labelled with a symbol $x$ from a finite alphabet $\Sigma$, each of which is associated with a…
forky40
- 7,988
- 2
- 12
- 33
3
votes
1 answer
Why can the max-relative entropy be written as $D_{\max}(\rho \parallel \sigma) = \inf \{ \lambda : \rho \leq 2^\lambda \sigma \}$?
The quantum conditional min-entropy is defined as
$$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B \right),$$
where
$$D_{\max}(\rho \parallel I_A \otimes \sigma_B) = \inf \{ \lambda : \rho \leq…
Josh
- 427
- 3
- 10
3
votes
1 answer
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 between $H_{\min(\max)}(A)_\rho$ and $H(A)_\rho$? Put…
Shadumu
- 373
- 1
- 5
3
votes
2 answers
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
$$\mathsf{H}_{\rm min }(\mathsf{X} \mid \mathsf{Y})_{\rho}\equiv -\inf _{\sigma \in \mathsf{D}(\mathcal Y)} \mathsf{D}_{\rm max…
glS
- 27,510
- 7
- 37
- 125
3
votes
1 answer
Continuity of Renyi entropies - limiting cases
The Renyi entropies are defined as
$$S_{\alpha}(\rho)=\frac{1}{1-\alpha} \log \operatorname{Tr}\left(\rho^{\alpha}\right), \alpha \in(0,1) \cup(1, \infty)$$
It is claimed that this quantity is continuous i.e. for $\rho, \sigma$ close in trace…
user1936752
- 3,311
- 1
- 9
- 24
2
votes
1 answer
Prove that for a cq-state $\rho_{XE}$, $H_\infty(X|E) \ge H_\infty(X) - \log|E|$
Given a classical-quantum(cq) state $\rho_{XE}$, where the $X$ register is classical, I want to prove the following:
$$
\begin{align}
H_\infty(X|E) \ge H_\infty(X) - \log|E|,\tag{1}
\end{align}
$$
i.e., the conditional min-entropy of $X$, having…
QuestionEverything
- 1,837
- 12
- 23
1
vote
1 answer
Does the max-relative entropy satisfy $0 < D_{\max}(\rho \parallel I_A \otimes \sigma_B) < 1$?
The quantum conditional min-entropy is defined as
$$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B \right)$$
where in general
$$D_{\max}(\rho \parallel \sigma) = \inf \{ \lambda : \rho \leq…
Josh
- 427
- 3
- 10
1
vote
1 answer
Prove that the conditional min-entropy is $H_{\rm min}(A|B)=\max_\sigma\sup\{\lambda:\,\rho\le 2^{-\lambda}(I\otimes\sigma)\}$
I have seen various definitions of quantum conditional min-entropy, which I believe are equivalent.
$$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B \right)$$
where in general
$$D_{\max}(\rho…
Josh
- 427
- 3
- 10
1
vote
1 answer
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 and this other related posts.
Given a bipartite…
glS
- 27,510
- 7
- 37
- 125