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7 votes

How does the conditional min-entropy $H_{\rm min}(A|B)_\rho$ relate to the conditional entropy $H(X|Y)_\rho$?

The conditional min-entropy $\text{H}_{\text{min}}(A | B)_{\rho}$ can be defined for an arbitrary state $\rho$ of a pair of registers $(A,B)$ as $$ - \inf_{\sigma} \,\text{D}_{\text{max}}(\rho \| \...
John Watrous's user avatar
  • 6,097
7 votes
Accepted

How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?

We can bound the amount of information that can be retrieved from $|\psi\rangle$ using Holevo's bound. Alice and Bob Let us first reformulate the situation in the terms usually employed in the context ...
Adam Zalcman's user avatar
  • 22.9k
4 votes
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Prove that for a cq-state $\rho_{XE}$, $H_\infty(X|E) \ge H_\infty(X) - \log|E|$

Let $\rho_{XE} = \sum_x p(x) |x\rangle \langle x| \otimes \rho_E(x)$ where $p(x)$ is a probability distribution and for each $x$, $\rho_E(x)$ is a quantum state on the system $E$. Let $\|X\|_1 = \...
Rammus's user avatar
  • 5,853
4 votes

What are explicit examples of smoothed conditional min(max) entropies?

I'll just give a classical example, which is a typical motivating example for these smooth quantities. Consider an $n$-bit distribution of the form $$ p(x) = \begin{cases} 1-\delta \qquad \text{ if }x=...
Rammus's user avatar
  • 5,853
4 votes
Accepted

What is the conditional min-entropy for diagonal ("classical") matrices?

Long story short: taking $\sigma_B = \rho_B$ is equivalent to taking the worst case min-entropy $$ \hat{H}_{\min}(A|B) = - \log \max_{a,b} P(A=a|B=b)\,, $$ and optimizing over $\sigma_B$ is equivalent ...
Rammus's user avatar
  • 5,853
3 votes
Accepted

What is the conditional min-entropy of a pure bipartite state?

I'll use an equivalent definition of the min-entropy $$ \begin{aligned} H_{\min}(A|B) = - \log_2 \min& \quad \lambda \\ \mathrm{s.t.}& \quad \rho_{AB} \leq \lambda I_A \otimes \sigma_B \\ &...
Rammus's user avatar
  • 5,853
3 votes
Accepted

Data Processing equality variation

No, the co-isometry map $\sigma \to V^\dagger \sigma V$ is not trace preserving. In the worst case you can have something like this. Take an isometry $V: \mathbb{C}^2 \rightarrow \mathbb{C}^3$ which ...
Rammus's user avatar
  • 5,853
3 votes
Accepted

Continuity of Renyi entropies - limiting cases

Assuming everything is finite dimensional. For $S_0$ we have $$S_0(\rho) = \log \mathrm{rank}(\rho).$$ It's pretty straightforward to see this is not continuous. Take $\rho_{\epsilon} = \epsilon |0\...
Rammus's user avatar
  • 5,853
3 votes

Is the quantum min-relative entropy $D_{\min}(\rho\|\sigma)=-\log(F(\rho, \sigma)^2)$ or $D_{\min}(\rho\|\sigma)=-\log(tr(\Pi_\rho\sigma))$?

As @rnva points out these are not the same quantities. To give some clarity as to why they are both referred to as $D_{\min}$ it is best to look at the as limiting cases of $\alpha$-R'enyi divergences....
Rammus's user avatar
  • 5,853
2 votes

How to calculate the conditional min-entropy via a semidefinite program?

I think I have an answer. The following should be the CVX code for one of the formulations found in this link. ...
QuestionEverything's user avatar
1 vote

What is the conditional min-entropy of a pure bipartite state?

Reading again the paper you linked, I think the way the authors were thinking about the result was of showing this via the relations between conditional min- and max-entropies, see discussion at the ...
glS's user avatar
  • 25.4k
1 vote

Which quantum entropies are meaningful with respect to continuous distributions of states?

I've found a partial answer for the case of conditional min-entropy, due to Ref. [1] (Appendix IV.B): Consider a fixed ensemble $\{(\rho_B(x), p(x))\}_{x \in \Sigma}$, where $p(x)$ is a probability ...
forky40's user avatar
  • 7,113
1 vote

Does the max-relative entropy satisfy $0 < D_{\max}(\rho \parallel I_A \otimes \sigma_B) < 1$?

No. As discussed e.g. in the second lecture of https://cs.uwaterloo.ca/~watrous/QIT-notes/, between pages 16 and 17, if $\sigma$ is a state, then $2^{-D_{\rm max}(\rho\|\sigma)}\in[0,1]$, or ...
glS's user avatar
  • 25.4k
1 vote

Why can the max-relative entropy be written as $D_{\max}(\rho \parallel \sigma) = \inf \{ \lambda : \rho \leq 2^\lambda \sigma \}$?

Assuming I'm reading the post correctly, the question seems to be "why/how is the standard notion of relative entropy related to the given expression for the max-relative entropy? Consider the ...
glS's user avatar
  • 25.4k
1 vote
Accepted

Prove that the conditional min-entropy is $H_{\rm min}(A|B)=\max_\sigma\sup\{\lambda:\,\rho\le 2^{-\lambda}(I\otimes\sigma)\}$

Observe that $$ H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \| I_A \otimes \sigma_B \right) = \sup\limits_{\sigma_B} [-D_{\max} \left( \rho_{AB} \| I_A \otimes \sigma_B \right)]...
glS's user avatar
  • 25.4k
1 vote
Accepted

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

I'm not sure what you mean exactly with "small gap", but you can easily build examples where $H(A)$ and $H_{\rm max}(A)$ are "maximally different". For example, $$\rho = \begin{...
glS's user avatar
  • 25.4k
1 vote

In what sense is the "conditional min-entropy" a conditional entropy?

Here is a perspective on why $H_{min}(A|B)$ is a min entropy, which may not directly answer your question. From an operational perspective, $H_{min}(A|B)$ is defined analogously to $H_{min}(A)$, at ...
forky40's user avatar
  • 7,113

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