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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
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How to derive the quantum Fisher information from the relative entropy?

Expressing the derivative $\partial_i\rho$ in terms of its eigenvalues and eigenvectors will show us that these two are not equal. I will assume a full-rank density matrix $\rho$ to streamline the ...
Quantum Mechanic's user avatar
3 votes

Clarification about inverses in sandwiched Renyi divergence

Firstly, the sandwiched divergence can be infinite even when $\rho$ and $\sigma$ are not orthogonal. For example, consider $\rho = \frac{|0\rangle \langle 0| + |1\rangle\langle 1|}{2}$ and $\sigma = |...
Rammus's user avatar
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Quasi concavity of max-relative entropy?

No, this is not possible. Consider $\rho_1 = \sigma_2 = \vert 0\rangle\langle 0 \vert$ and $\rho_2 = \sigma_1 = \vert 1\rangle\langle 1 \vert$. Then, $$D_{\max}(\rho_i\|\sigma_i) = \infty\quad \text{...
rnva's user avatar
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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
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What can be said about the non-negativity of the relative entropy of $S(\rho_{AB}||\rho_{B})$?

Source of the problem The purported contradiction arises due to the use of incorrect assumptions for Klein equality $$ S(\rho||\sigma) \ge 0. $$ The inequality does not require any particular ...
Adam Zalcman's user avatar
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2 votes
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When can the max relative entropy be written as $D_{\max}(\rho\|\sigma) = \|\sigma^{-1/2}\rho\sigma^{-1/2}\|_{\infty}$?

There is a problem in the derivation you presented, since $\rho \leq \lambda \sigma$ is only equivalent to $\sigma^{-1/2} \rho \sigma^{-1/2} \leq \lambda I$ when $\sigma$ is invertible (or at least ...
user13507's user avatar
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Questions about the relation between max-relative entropy $D_{\max}(\rho||\sigma)$ and max-information

Can someone provide an example of a state $\rho_{AB}$ for which $\sigma^\star_B \neq \rho_B$? Why not start very easily, with a separable state such as $$ \rho_{AB}=\left(p_0|0\rangle\langle 0|\...
DaftWullie's user avatar
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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
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Showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for classical-quantum states

As you say, $$ \mathrm{Tr}[\rho_{XB} \log \rho_{XB}] = -S(X) + \sum_{x} p(x) \mathrm{Tr}[\rho_{B}^x \log \rho_B^x]. $$ But if you can prove the above statement, then the exact same derivation gives ...
Rammus's user avatar
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2 votes

How to take the limits of the sandwiched Renyi divergences?

Both limits are dealt with in a fair amount of detail in the work that originally defined the sandwiched entropies: On quantum Renyi entropies: a new generalization and some properties. In particular, ...
Rammus's user avatar
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Quantum Relative entropy- the math and intuition

To answer your questions briefly: This is purely a matter of normalization. It doesn't matter really as you could define it for just positive semidefinite operators like Watrous and they will be the ...
Rammus's user avatar
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Quantum relative entropy between pre- and post-measurement states

Firstly notice that the divergences are $$ D(\rho\|\rho_{\mathrm{rank-one}}) = - S(\rho) + S(\rho_{\mathrm{rank-one}}) $$ and $$ D(\rho\|\rho_{\mathrm{coarse}}) = - S(\rho) + S(\rho_{\mathrm{coarse}}) ...
Rammus's user avatar
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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
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1 vote
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Calculation of $\frac{d}{dt} I_t(A,X)$ in proving the convexity of the relative entropy via Lieb's theorem

By linearity, $$\partial_t \operatorname{tr}(F(t))=\operatorname{tr}(F'(t)).$$ Then, $$\partial_t A^t=\partial_t e^{t\log A}= \log(A) A^t,$$ $$\partial_t \operatorname{tr}(B A^t C A^{1-t}) = \...
glS's user avatar
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1 vote

Does the quantum relative entropy have a direct operational interpretation?

I would argue yes, in the context of recoverability. Given a quantum channel $\mathcal{N}: L(A) \to L(B)$ and a state $\sigma$ on $A$, we say that a quantum channel $\mathcal R: L(B) \to L(A)$ is a $(...
Rammus's user avatar
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1 vote
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Conditional entropy as relative entropy between probability distributions

The relative entropy is defined as $$ D(p\|q) = \sum_x p(x) \log\left(\frac{p(x)}{q(x)}\right). $$ The conditional entropy of $Y$ given $X$ for two random variables $X$ and $Y$ is defined as $$ H(Y|X) ...
Rammus's user avatar
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1 vote

What is the quantum relative entropy between pure states?

Well, if you go back to the definition of quantum relative entropy you'll notice it is not defined for all states. Or more precisely, it is defined as $$S(\rho\|\sigma) = \operatorname{Tr}(\rho\log\...
glS's user avatar
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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
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1 vote

Data processing inequality for relative entropy in the presence of an amplitude damping channel

An easy (but perhaps cheating) answer: The relative entropy $S(A||B)$diverges when $A$ has support over the kernel of $B$ (e.g., wikipedia). Now, the kernel of $B=|0\rangle^{\otimes n}\langle 0|^{\...
Quantum Mechanic's user avatar

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