Questions tagged [max-entropy]

For questions about the max entropy (also called Hartley entropy) or max-relative entropy functions.

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Minimising the largest max-relatve entropy any quantum state can have with respect to other states drawn from a set

The max-relative entropy between two states is defined as $D_{\max}(\rho||\sigma) := -\log \max\{\lambda: \lambda\rho \le \sigma \}$. Let $\mathcal{R} := \{\rho_k\}$ be a set of $K$ states for a ...
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2 votes
1 answer
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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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2 votes
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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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Why are "smooth entropic quantities" useful/necessary?

Consider the $\epsilon$-smoothed relative max-entropy of $\rho$ with respect to $Q$, defined as (following Watrous' notation from these notes): $$\mathrm D_{\rm max}^{\epsilon}(\rho\|Q) = \min_{\xi\in ...
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3 votes
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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. ...
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3 votes
1 answer
47 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. ...
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3 votes
1 answer
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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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3 votes
1 answer
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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-\...
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2 votes
1 answer
76 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 $A\leq B$ ...
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2 votes
1 answer
134 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
1 answer
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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 ...
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4 votes
1 answer
73 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 ...
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1 vote
1 answer
77 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-...
5 votes
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130 views

Connection between smooth max-relative entropy and smooth 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 ...
  • 2,301
2 votes
1 answer
132 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 ...
  • 2,301
4 votes
1 answer
117 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 ...
  • 2,301