5
votes
Accepted
What are non-standard ways to describe the distance between states?
One nonstandard approach to measuring distances between states is the "$\varepsilon$-smooth relative complexity distance" $\mathcal C_\varepsilon(|\psi\rangle,|\phi\rangle)$ corresponding to ...
- 13.6k
5
votes
Accepted
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 ...
- 4,569
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 ...
- 6,226
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 = |...
- 6,226
3
votes
Accepted
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,
$$D_{\max}(\rho_i\|\sigma_i) = \infty\quad \text{...
3
votes
Is the quantum min-relative entropy $D_{\min}(\rho\|\sigma)=-\log(F(\rho, \sigma)^2)$ or $D_{\min}(\rho\|\sigma)=-\log(tr(\Pi_\rho\sigma))$?
As @rnva points out these are not the same quantities. To give some clarity as to why they are both referred to as $D_{\min}$ it is best to look at the as limiting cases of $\alpha$-R'enyi divergences....
- 6,226
3
votes
Accepted
What can be said about the non-negativity of the relative entropy of $S(\rho_{AB}||\rho_{B})$?
Source of the problem
The purported contradiction arises due to the use of incorrect assumptions for Klein equality
$$
S(\rho||\sigma) \ge 0.
$$
The inequality does not require any particular ...
- 24.2k
2
votes
Exercise 11.7 in Nielsen & Chuang and basic properties of Shannon entropy
It's just a thought experiment inspired by John Watrous's deleted comment.
Suppose we have a probability distribution $p(x,y)$ and we want to approximate it with $q(x,y)$. Suppose that we know $p(x)$, ...
- 3,181
2
votes
Accepted
Quantum relative entropy between pre- and post-measurement states
Firstly notice that the divergences are
$$
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}})
...
- 6,226
2
votes
Clarification about inverses in sandwiched Renyi divergence
This quantity was defined independently in https://arxiv.org/abs/1306.1586, where many of its properties were explored therein.
To answer your question, the quantity can be defined in the general case ...
- 121
2
votes
Accepted
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 ...
- 126
2
votes
Accepted
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
$$
\rho_{AB}=\left(p_0|0\rangle\langle 0|\...
- 60.7k
2
votes
How to calculate the conditional min-entropy via a semidefinite program?
I think I have an answer. The following should be the CVX code for one of the formulations found in this link.
...
- 1,785
2
votes
Accepted
Showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for classical-quantum states
As you say,
$$
\mathrm{Tr}[\rho_{XB} \log \rho_{XB}] = -S(X) + \sum_{x} p(x) \mathrm{Tr}[\rho_{B}^x \log \rho_B^x].
$$
But if you can prove the above statement, then the exact same derivation gives ...
- 6,226
2
votes
How to take the limits of the sandwiched Renyi divergences?
Both limits are dealt with in a fair amount of detail in the work that originally defined the sandwiched entropies: On quantum Renyi entropies: a new generalization and some properties. In particular, ...
- 6,226
2
votes
Accepted
Quantum Relative entropy- the math and intuition
To answer your questions briefly:
This is purely a matter of normalization. It doesn't matter really as you could define it for just positive semidefinite operators like Watrous and they will be the ...
- 6,226
1
vote
Which quantum entropies are meaningful with respect to continuous distributions of states?
I've found a partial answer for the case of conditional min-entropy, due to Ref. [1] (Appendix IV.B):
Consider a fixed ensemble $\{(\rho_B(x), p(x))\}_{x \in \Sigma}$, where $p(x)$ is a probability ...
- 7,448
1
vote
Accepted
Calculation of $\frac{d}{dt} I_t(A,X)$ in proving the convexity of the relative entropy via Lieb's theorem
By linearity,
$$\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♦
- 26.3k
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 $(...
- 6,226
1
vote
Accepted
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
$$
H(Y|X) ...
- 6,226
1
vote
What is the quantum relative entropy between pure states?
Well, if you go back to the definition of quantum relative entropy you'll notice it is not defined for all states. Or more precisely, it is defined as
$$S(\rho\|\sigma) = \operatorname{Tr}(\rho\log\...
glS♦
- 26.3k
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 ...
- 7,448
1
vote
What is the conditional min-entropy for diagonal ("classical") matrices?
Classical definition of $\mathsf D_{\rm max}(P\|Q)$
$\newcommand{\H}{\mathsf{H}}\newcommand{\Hmin}{\H_{\rm min}}\newcommand{\D}{\mathsf{D}}\newcommand{\Dmax}{\D_{\rm max}}$Consider the max-relative ...
glS♦
- 26.3k
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|^{\...
- 4,569
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