All Questions
Tagged with textbook-and-exercises information-theory
16
questions
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0
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65
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Distinguish two states with their priors probability
EDIT: This is a computer programming / coding exercise
The states $\left|\psi\right>$ and $\left|\phi\right>$ are defined as
$∣ϕ⟩=\cos(θ_ϕ)\left|0\right>+\sin(θ_ϕ) \left|1\right>$ with ...
- 123
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
views
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, ...
- 831
1
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1
answer
138
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In Schumacher’s noiseless channel coding theorem, how do we get the exponents in $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\otimes n(1−p)/2}$?
On pg. 55 in Nielsen and Chuang, it's said that:
the $|0\rangle + |1\rangle$ product can be well approximated by a superposition of states of the form $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\...
- 669
5
votes
1
answer
513
views
Can all mixed states be written as a convex combination $\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|$?
States belonging to some space $\mathcal H$ can be described by density operators $\rho\in L(\mathcal H)$ that are positive and have trace one. Pure states are the ones that can be written as $\rho=|\...
- 63
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1
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50
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Why do we want the no error limit to be 1?
In a textbook by Nielsen and Chuang, there's the following paragraph:
The idea of quantum data compression is that the compressed data should be recovered with very good fidelity. Think of the ...
- 669
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
votes
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
2
votes
1
answer
138
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In classical state discrimination, why does the trace distance quantify the probability of success?
Consider the following task: we are given a probability distribution $p_y:x\mapsto p_y(x)$ with $y\in\{0,1\}$ (e.g. we are given some black box that we can use to draw samples from either $p_0$ or $...
glS♦
- 25.9k
5
votes
0
answers
95
views
Pure state ensembles achieving the Holevo $\chi$-quantity with at most $d^2$ states
Theorem 8.10 in Chapter 8 of Theory of Quantum Information asserts that the Holevo capacity of a quantum channel (between density operators on $\mathbb{C}^d$) can be achieved by an ensemble consisting ...
- 2,111
4
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0
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80
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A question in classical and quantum information
Let $\rho, \sigma \in \mathfrak{D}(A)$ with $\operatorname{supp}(\rho) \subseteq \operatorname{supp}(\sigma),$ and spectral decomposition
$$
\rho=\sum_{x} p_{x}\left|\psi_{x}\right\rangle\left\langle\...
- 133
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}
...
5
votes
2
answers
713
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Nielsen and Chuang ex 2.73
I've been trying to solve exercise 2.73 (p.g 105), and I'm not sure if i'v been overthinking it and the answer is as simple as i've described below or if I am missing something, or i'm just wrong!
Ex ...
- 969
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