Questions tagged [entropy]
For questions about the various kinds of entropies --- as defined in the context of quantum information theory and quantum statistical mechanics.
31
questions with no upvoted or accepted answers
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34
views
Relative entropy inequality for many copies of a channel
Suppose we have two quantum channels $\mathcal{E}_{A\rightarrow B}, \mathcal{F}_{A\rightarrow B}$ that satisfy
$$D(\mathcal{E}(\rho_A)\|\mathcal{E}(\sigma_A))\geq D(\mathcal{F}(\rho_A)\|\mathcal{F}(\...
- 51
5
votes
0
answers
144
views
Connection between smooth max-relative entropy and smooth max-information
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
- 2,597
5
votes
0
answers
270
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Schumacher compression - comparing with Shannon compression
Background
Shannon's source coding theorem tells us the following. We shall consider a binary alphabet for simplicity. Suppose Alice has $n$ independent and identically distributed instances of a ...
- 2,597
4
votes
0
answers
41
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Is there a known deterministic counterexample for non-additivity of minimal output entropy?
Hastings has proved that the minimal output entropy is not additive: it may happen that $S_{\mathrm{min}}(\Phi_1 \otimes \Phi_2) < S_{\mathrm{min}}(\Phi_1)+S_{\mathrm{min}}(\Phi_2) $ for quantum ...
- 141
4
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39
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Does the quantum relative entropy have a direct operational interpretation?
Consider the quantum relative entropy, defined as
$$D(\rho\|\sigma) = \operatorname{tr}(\rho\log\rho)-\operatorname{tr}(\rho\log\sigma),$$
for all $\rho,\sigma\ge0$ such that $\operatorname{im}(\rho)\...
glS♦
- 23.4k
4
votes
0
answers
243
views
Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy
The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is
$$
S(A,B)\ge|S(A)-S(B)|
$$
this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
- 791
4
votes
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answers
79
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Proof of upper and lower bound (Gilbert-Varshamov bound) for linear code
I am trying to prove the following bounds for a $[n, k]$ code that can correct $t$ errors
\begin{align}
1-H\left(\frac{t}{n}\right)\geq \frac{k}{n}\geq 1-H\left(\frac{2t}{n}\right)
\end{align}
where
\...
- 920
4
votes
0
answers
33
views
Max-relative entropy quasi-convexity inequality under partial trace
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$
It is known that the max-relative entropy is quasi-convex. ...
- 2,597
4
votes
0
answers
116
views
Why does the entanglement entropy give the number of singlets required to create a given state?
I've read that, given a bipartite pure state $|\Phi\rangle$, its entanglement (equivalently here, von Neumann) entropy $E(\Phi)$ gives the asymptotic number of singlets required to create $n$ copies ...
- 145
4
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132
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Calculating the entropy of a quantum state
Let $\rho_{AR}$ be some $d-$dimensional pure quantum state. Consider a channel $N_{A\rightarrow B}$ that outputs a constant state in $B$. We now consider the Stinespring dilation of this channel with ...
- 2,597
4
votes
0
answers
100
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Forbidden/allowed outputs of a quantum channel
The coherent information of a channel $\mathcal{E}_{A'\rightarrow B}$ is defined as the maximum value obtained by the following function where the maximization is over all input states
$$I_{\rm{coh}}(...
- 2,597
3
votes
0
answers
144
views
Increasing the von Neumann entropy despite the measurement?
Background
Assume we have a density matrix $\rho$ of a sub-ensemble. However, we have an imperfect measuring instrument. While it does perform a measurement, we do not know exactly when it performs ...
- 429
3
votes
0
answers
133
views
Convexity of coherent information - erroneous argument!
Consider a state $\rho_{AB}$. Let it have purification $\psi_{A'AB}$. I am interested in the coherent information of this state which is given by
$$I(A\rangle B)_\rho = S(B)_\rho - S(AB)_\rho$$
I ...
- 2,597
2
votes
0
answers
47
views
General structure of the state with $I(A:B|C)_{\rho}{=}2 \log_2 \{\min (d_A, d_B)\}$
The conditional quantum mutual information (CQMI) of a state $\rho^{ABC}$ respects the dimension bound $I(A:B|C)_{\rho}{\leq}2 \log_2 \{\min (d_A, d_B)\}$ (Mark Wilde's book, exercise 11.7.9). One ...
- 135
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answers
56
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Can't understand how $D_{A,B} + D_{A,C} \ge E_{A,B}+E_{A,C}$ is proved used the subadditivity of entropy
I am reading Monogamy properties of quantum and classical correlations. Eq.10 states that $$D_{A,B} + D_{A,C} \ge E_{A,B}+E_{A,C},$$ where $D_{i,j}$ is the quantum discord, and $E_{A,B}$ is the ...
- 1,115
2
votes
0
answers
41
views
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):
$$\mathrm D_{\rm max}^{\epsilon}(\rho\|Q) = \min_{\xi\in ...
glS♦
- 23.4k
2
votes
0
answers
492
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What does a quantum mutual information larger than its classical upper bound represent?
Let $\rho$ be a bipartite state. Its quantum mutual information is defined as
$$\newcommand{\tr}{\operatorname{tr}}I(\rho) = S(\tr_B(\rho)) + S(\tr_A(\rho)) - S(\rho),$$
where $S(\sigma)$ is the von ...
glS♦
- 23.4k
2
votes
0
answers
355
views
How to prove that the mutual information is subadditive?
Let $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$ be two vectors of random variables. To make things concrete, assume that Alice sends each component $x_j$ through a noisy channel to Bob, ...
- 165
2
votes
0
answers
109
views
Partial trace instead of trace in definition of entropy
For a bipartite quantum state $\rho_{AB}$, we have that the von Neumann entropy is
$$S(\rho_{AB}) = -\text{Tr}(\rho_{AB}\log\rho_{AB})$$
If instead, one took the partial trace above and obtained
$$\...
- 2,597
2
votes
0
answers
45
views
Quantum analogues of information theoretic measures: are log probabilities replaced with the density matrix?
Below is a question and an answer.
How does quantum information relate to, diverge from or reduce to
Shannon information, which used log probabilities?
What people are more often interested in are ...
- 947
2
votes
0
answers
264
views
Conditional Time Evolution increases entropy?
Question
Does the below calculation conclusively show the idea of conditional time evolution (if state measured is $x$ I do $y$ else I do $z$ ) increases the Von Neumann entropy? Has this already ...
- 429
1
vote
0
answers
39
views
Why are all Rényi entropies equal for Clifford dynamics?
In this paper, by Adam Nahum et al., the authors trivially states that "For Clifford dynamics all Rényi entropies are equal ... " which is not trivial to me.
Is there a paper or lecture ...
1
vote
0
answers
22
views
States for Tight Maassen-Uffink Uncertainty Relation
I was reading this paper titled
"Entropic Uncertainty Relations and their Applications". There,at equation (47) we have the Maassen-Uffink uncertainty relation which states that for a pair ...
- 1,775
1
vote
0
answers
29
views
Find the minimal and maximal of $\widehat{S}_f (\rho \| U^* \sigma U)$
I have been study the minimal (maximal) of a $f-$divergence. Fumio Hiai introduced the $\widehat{S}_f (\rho \| \sigma)$ divergence in his article.
$$\widehat{S}_f (\rho \| \sigma) := \text{Tr} \sigma^{...
- 111
1
vote
0
answers
64
views
von Neumann entropy of an arbitrary composition system
I understand the von Neumann entropy of a $2$-composite system is that of the reduced density matrix? What is the von Neumann entropy of an entanglement of a more than $3$ composite system?
- 217
1
vote
0
answers
16
views
What is the idea behind compressibility results in terms of Renyi entropies?
In (Tomamichel 2015), in (1.2) the author mentions the result that a source $X$ with probability distribution $\rho\equiv\rho_X$ admits an $(\varepsilon,m)$-code as long as there is some $\alpha\in[\...
glS♦
- 23.4k
1
vote
0
answers
42
views
Understanding conditional $L_2$ distances
I see that conditional $L_2$ distances from uniform are defined in the following way:
$L_2(\rho_{AB}\vert \sigma_B)= \text{tr}\left(((\rho_{AB}- \mu_{A} \otimes \rho_{B}) (\mathbb{I}_A \otimes \...
- 469
1
vote
0
answers
66
views
Linear and Logarithmic Constraint in Semidefinite Programming
I am trying to minimize the largest component of a vector $x = [x_1, x_2, x_3, x_4]$, where $x_1 \ge x_2 ... \ge x_4$, such that it satisfies a set of linear inequalities $A, b$ in the following way:
$...
- 1,775
1
vote
0
answers
164
views
Continuity of relative entropy variance
Related question here - copying over the definitions.
The relative entropy between two quantum states is given by $D(\rho\|\sigma) = \operatorname{Tr}(\rho\log\rho -\rho\log\sigma)$. It is known that ...
- 2,597
0
votes
0
answers
57
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Proof of the Lieb's theorem
Lemma A6.2: Let $R1 , R2 , S1 , S2 , T1, T2$ be positive operators such that
$0 = [R1, R2 ] = [S1, S 2 ] = [T1, T2 ]$, and
$$
R1 ≥ S1 + T1\\
R2 ≥ S2 + T2
$$
Then for all $0 ≤ t ≤ 1$,
$$
R_1^t R_2^{1−t}...
- 791
0
votes
0
answers
146
views
Coherence measurement for density matrix
I have a density matrix of the form:
$$\rho(t)=\left[
\begin{array}{ccc}
\frac{1}{4}+\frac{1}{12} e^{-\frac{2 \tau ^{2 H+2}}{2 H+2}} & \frac{1}{3} & \frac{1}{4}+\frac{1}{12} e^{-\frac{2 \tau ^...
- 11