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.
18
questions
3
votes
3
answers
111
views
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
...
- 55
1
vote
0
answers
21
views
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 ...
- 6,143
1
vote
1
answer
29
views
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
$$D_{\max}(\rho \...
- 365
3
votes
1
answer
71
views
Why can the max-relative entropy be written as $D_{\max}(\rho \parallel \sigma) = \inf \{ \lambda : \rho \leq 2^\lambda \sigma \}$?
The quantum conditional min-entropy is defined as
$$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 \...
- 365
1
vote
1
answer
45
views
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.
$$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B ...
- 365
3
votes
1
answer
106
views
Difference between min/max-entropies and the von Neumann entropy
Consider the (smooth) min-entropy, max-entropy and von Neumann entropy of a given density operator $\rho_A$. Does a small gap between $H_{\max(\min)}(A)_\rho$ and $H(A)_\rho$ implies a small gap ...
- 279
3
votes
1
answer
173
views
What are explicit examples of smoothed conditional min(max) entropies?
Some general discussion of smoothed entropic quantities is found for example in Watrous notes, and an overview and discussion on its operational interpretations in (Koenig et al. 2008). It seems the ...
glS♦
- 23.4k
1
vote
1
answer
96
views
In what sense is the "conditional min-entropy" a conditional entropy?
$\newcommand{\H}{\mathsf{H}}\newcommand{\Hmin}{\H_{\rm min}}\newcommand{\D}{\mathsf{D}}\newcommand{\Dmax}{\D_{\rm max}}$Consider the conditional min-entropy $\Hmin(A|B)_\rho$, discussed e.g. in this ...
glS♦
- 23.4k
2
votes
2
answers
351
views
What is the conditional min-entropy for diagonal ("classical") matrices?
The conditional min-entropy, discussed e.g. in these notes by Watrous, as well as in this other post, can be defined as
$$\mathsf{H}_{\rm min }(\mathsf{X} \mid \mathsf{Y})_{\rho}\equiv -\inf _{\sigma \...
glS♦
- 23.4k
1
vote
0
answers
43
views
Understanding conditional $L_2$ distances
I see that conditional $L_2$ distances from uniform are defined in the following way:
$L_2(\rho_{AB}\vert \sigma_B)= \text{tr}\left(((\rho_{AB}- \mu_{A} \otimes \rho_{B}) (\mathbb{I}_A \otimes \...
- 469
1
vote
0
answers
62
views
Entropy relations in classical-quantum states with conditioned independence
Consider a classical-quantum (pure) state $\rho_{AEBC}$ where $A,C$ are classical registers.
Suppose $\rho$ can be written in the following form:
$$\rho = \sum_{c} \alpha_c \vert c\rangle\langle c|\...
- 469
1
vote
1
answer
65
views
Data Processing equality variation
Let $\rho_{AB}$ be a state and $T: B \rightarrow C$ be a CPTP map with $\sigma_{AC}= T(\rho_{AB})$.
It is well known that $H_{\infty}(A \vert B)_{\rho} \geq H_{\infty}(A \vert C)_{\sigma}$ (aka data ...
- 469
4
votes
1
answer
145
views
How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?
I create a quantum state $| \psi \rangle = \frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ for a randomly chosen $x_0,x_1$ of 50 bits. I give this quantum state $|\psi \rangle$ to you and you return ...
- 103
2
votes
1
answer
165
views
Prove that for a cq-state $\rho_{XE}$, $H_\infty(X|E) \ge H_\infty(X) - \log|E|$
Given a classical-quantum(cq) state $\rho_{XE}$, where the $X$ register is classical, I want to prove the following:
$$
\begin{align}
H_\infty(X|E) \ge H_\infty(X) - \log|E|,\tag{1}
\end{align}
$$
i.e....
- 1,775
3
votes
1
answer
139
views
Continuity of Renyi entropies - limiting cases
The Renyi entropies are defined as
$$S_{\alpha}(\rho)=\frac{1}{1-\alpha} \log \operatorname{Tr}\left(\rho^{\alpha}\right), \alpha \in(0,1) \cup(1, \infty)$$
It is claimed that this quantity is ...
- 2,597
5
votes
1
answer
153
views
Is the quantum min-relative entropy $D_{\min}(\rho\|\sigma)=-\log(F(\rho, \sigma)^2)$ or $D_{\min}(\rho\|\sigma)=-\log(tr(\Pi_\rho\sigma))$?
In John Watrous' lectures, he defines the quantum min-relative entropy as
$$D_{\min}(\rho\|\sigma) = -\log(F(\rho, \sigma)^2),$$
where $F(\rho,\sigma) = tr(\sqrt{\rho\sigma})$. Here, I use this ...
- 51
5
votes
1
answer
401
views
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, ...
- 1,775
8
votes
1
answer
547
views
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 $\...
- 81