Questions tagged [max-entropy]
For questions about the max entropy (also called Hartley entropy) or max-relative entropy functions.
11
questions
2
votes
0answers
17 views
Max-relative entropy quasi-convexity inequality under partial trace
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$
It is known that the max-relative entropy is quasi-convex. ...
3
votes
1answer
34 views
Quasi concavity of max-relative entropy?
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$
It is known that the max-relative entropy is quasi-convex. ...
3
votes
1answer
48 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 ...
3
votes
1answer
46 views
Do we know the limits of the quantum Tsallis entropy?
From the two main generalizations of the von Neumann entropy:
\begin{equation}
S(\rho)=-\operatorname{Tr}(\rho \log \rho)
\end{equation}
meaning Rényi:
\begin{equation}
R_{\alpha}(\rho)=\frac{1}{1-\...
2
votes
1answer
56 views
When can the max relative entropy be written as $D_{\max}(\rho\|\sigma) = \|\sigma^{-1/2}\rho\sigma^{-1/2}\|_{\infty}$?
The max-relative entropy between two states is defined as $D_{\max}(\rho\|\sigma) = \log\lambda$, where $\lambda$ is the smallest real number that satisfies $\rho\leq \lambda\sigma$, where $A\leq B$ ...
2
votes
1answer
82 views
Relating quantum max-relative entropy to classical maximum entropy
The quantum 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 - \...
2
votes
1answer
35 views
What is the relationship between these two definitions for the max-entropy?
On Wikipedia, the max-entropy for classical systems is defined as
$$H_{0}(A)_{\rho}=\log \operatorname{rank}\left(\rho_{A}\right)$$
The term max-entropy in quantum information is reserved for the ...
4
votes
1answer
68 views
Non-lockability of quantum max-entropy
Lockability and non-lockability are explained in this paper. A real valued function of a quantum state is called non-lockable if its value does not change by too much after discarding a subsystem. The ...
1
vote
1answer
56 views
Semi-definite program for conditional smooth max-entropy
I am aware of a SDP formulation for smooth min-entropy: question link. That program for smooth min-entropy was found in this book by Tomachiel: page 91. However, I am yet to come across a semi-...
2
votes
1answer
108 views
Continuity bounds on $D_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B)$
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 ...
4
votes
1answer
98 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 ...