All Questions
Tagged with textbook-and-exercises entropy
15
questions
2
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2
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161
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Exercise 11.7 in Nielsen & Chuang and basic properties of Shannon entropy
I apologize in advance if this question is trivial, I'm aware I'm a total beginner in this field. This is the exercise I would like to solve:
As to the first point, what I get is that one should ...
- 61
2
votes
2
answers
351
views
Derive the Concavity of Quantum Conditional Entropy from Strong subadditivity
In Exercise 11.25, Page 522, Entropy and information, Quantum Computation and Quantum Information by Nielsen and Chuang, it is required to show that the concavity of the conditional entropy may be ...
- 831
4
votes
0
answers
368
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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 ...
- 831
2
votes
2
answers
208
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How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?
The proof of the Fannes' inequality replies on the formula $T(ρ, σ)≥\sum_i|r_i − s_i|$, where $r_i,s_i$ are the eigenvalues of $\rho,\sigma$, in the descending order.
In the proof given in Box 11.2, ...
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1
vote
1
answer
279
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Conditional entropy as relative entropy between probability distributions
Find the expression for the conditional entropy $H(Y|X)$ as a relative entropy between two probability distributions. Use this expression to deduce that $H(Y |X)≥0$, and to find the equality ...
- 831
2
votes
2
answers
402
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How to understand intuitively the concavity of the binary entropy?
In Nielsen and Chuang's Quantum Computation and Quantum Information book, introducing the binary entropy, they gave an intuitive example about why binary entropy is concave:
Alice has in her ...
- 695
2
votes
1
answer
492
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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, ...
- 163
2
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1
answer
147
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Understanding the definition of entropy in the joint entropy theorem derivation
From section 11.3.2 of Nielsen & Chuang:
(4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|...
- 21
1
vote
1
answer
1k
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Calculate the von Neumann Entropy of a two-qubit entangled state
After working through an exercise I got a confusion answer/solution that either may be because I've made a mistake or I'm not understanding von Neumann Entropy.
I have the two qubit system
$$ | \psi \...
- 39
2
votes
1
answer
197
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von Neumann entropy in a limiting case
I am stuck with a question from the book Quantum theory by Asher Peres.
Excercise (9.11):
Three different preparation procedures of a spin 1/2 particle are represented by the vectors $\begin{pmatrix}
...
2
votes
1
answer
85
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How is $S(\rho)=H(p_{i})+\sum_{i}p_{i}S(\rho_{i})\le \log(d)$ possible if $\rho_{i}$ are not pure states?
I know how this can be proved using the quantum relative entropy. However, even with this proof, and am still confused about how this emerges.
Say I have a source that produces two states $\rho_1$ and ...
- 1,117
1
vote
1
answer
132
views
Proof of quantum data processing inequality in N&C on pg 566
On page 566, it states that using $S(\rho^{'})-S(\rho,\varepsilon) \ge S(\rho)$ and combining this with $S(\rho) \ge S(\rho^{'})-S(\rho,\varepsilon))$, we get $S(\rho^{'})=S(\rho)-S(\rho,\varepsilon)$....
- 1,117
2
votes
1
answer
170
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Conditional version of the triangle inequality for Von Neumann entropy
I'm trying to solve problem 11.3 in Nielsen Chuang:
(3) Prove the conditional version of the triangle inequality:
$$
S(A,B|C)\geq S(A|C)-S(B|C)
$$
But the inequality seems incorrect. For example,...
- 23
4
votes
2
answers
299
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In the proof of the joint entropy theorem, why are $p_i\lambda_i^j$ the eigenvalues?
From section 11.3.2 of Nielsen & Chuang:
(4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|...
- 535
1
vote
1
answer
4k
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How do I calculate the von Neumann entropy of a pure one-qubit density matrix?
Let's say I have a pure state of the form:
$$\psi = \sqrt{\frac{3}{9}} \lvert 0 \rangle + \sqrt{\frac{6}{9}} \lvert 1 \rangle$$
Then the density matrix representation would be:
$$\rho = \psi \otimes \...
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