Questions tagged [nielsen-and-chuang]
For questions about exercises or passages from the popular quantum computing textbook *Quantum Computation and Quantum Information* by Michael Nielsen and Isaac Chuang.
323
questions
1
vote
2
answers
59
views
Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
I have trouble understanding this proof in Nielsen & Chuang, specifically the identity in $(10.20)$, which reads $$ U_k^\dagger P_k F_l \sqrt{\rho} = U_k^\dagger P_k^\dagger P_k^\dagger F_l P \...
- 13
5
votes
1
answer
105
views
Question about Nielson & Chuang Problem 9.2
I am working on the following problem from the book "Quantum Computation and Quantum Information" by Nielsen and Chuang.
Problem 9.2: Let $\mathcal{E}$ be a trace-preserving quantum
...
- 53
2
votes
0
answers
59
views
Quantum Process Tomography for 2 qubits
I need clarification on a few aspects related to Box 8.5 and Exercise 8.34 from the book Quantum Computation and Quantum Information by Nielsen & Chuang .
While attempting Exercise 8.34, I ...
2
votes
2
answers
54
views
What we get when measure $|0\rangle$ under computational basis?
It is said if we have been given the state $|0\rangle$, the measurement will yield $0$ with probability $1$ in Nielsen's book.
So here, the measurement will yield $0$ refers to we will get state $|0\...
- 657
0
votes
2
answers
76
views
Explanation of the 2.60 equation page 76 in the Nielsen and Chuang [duplicate]
In the Nielsen and Chuang book page 76, equation 2.60 says that we can rewrite the trace $$Tr(A \left|\psi\right>\left<\psi\right|)$$ as follow :
$$Tr(A \left|\psi\right>\left<\psi\right|) ...
- 57
0
votes
1
answer
160
views
Unital channel which is not mixed unitary
How to prove that for a multi-qubit system a unital channel is not necessarily mixed unitary? This is Problem 8.3 in Nielsen and Chuang. Here's a snippet of the text:
Shall I need to take two ...
3
votes
1
answer
148
views
Proof of the 4.11 exercise in the Nielsen & Chuang book
In question 4.11 in Nielsen and Chuang's book, it states that there is a formula to describe any unitary matrix $U$ with two vectors $\vec{n}$ and $\vec{m}$ in the following way:
$$U=\exp(i \alpha) ...
- 57
1
vote
3
answers
117
views
Exercise 4.16 in the Nielsen & Chuang book
In the 4.16 exercice in the Quantum Computation and Quantum Information (Michael A. Nielsen & Isaac L. Chuang), I don't understand why the correct answer is not this matrix :
$$
\left[ {\begin{...
- 57
1
vote
1
answer
44
views
Unitarity of a matrix in the EPR experiment
I'm having troubles in understanding a statement in Box 2.7 at page 113 in the Nielsen & Chuang.
Firstly, it assumed to be working with a two-qubits quantum system in state $|\psi\rangle = \frac{|...
- 11
1
vote
1
answer
89
views
Can a density operator be written equivalently as $\rho=\sum_i p_i|\psi_i〉\!\langle\psi_i|$ and $\rho=\sum_i\lambda_i|\psi_i\rangle\!\langle\psi_i|$?
My doubt arises from page 99, 101 of the book Quantum Computation and Quantum Information by Michael A.Nielson and Issac L.Chung.
Let {${p_{i}, | \psi_{i} \rangle }$} be an ensemble of pure states.
...
- 510
4
votes
2
answers
141
views
How to derive the expression for the probability in quantum phase estimation? ((5.27) Nielsen & Chuang)
I'm trying to understand the QPE algorithm that is presented in the
Nielsen and Chuang textbook. More precisely, I do not understand Equation $(5.27)$.
Context: In the following, let $b$ be a natural ...
- 41
2
votes
0
answers
64
views
$T_1$ and $T_2$ time with amplitude damping
Exercise 8.30 of Nielson & Chuang's QCQI says
Equation 7.144, which is mentioned in the text, is
$$\begin{bmatrix}
a & b\\
b^* & 1-a
\end{bmatrix}\rightarrow\begin{bmatrix}
(a-a_0)e^{-t/...
- 53
0
votes
0
answers
93
views
Show that for pure states the description of the Bloch vector we have given coincides with that in section 1.2
$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} $
I am having a little bit of difficulty with part (4) of Exercises 2.72 from Nielsen and Chuang's "Quantum ...
- 295
5
votes
2
answers
262
views
Why are rotations represented by exponentials of Pauli matrices?
I'm self-studying Quantum Computation from Nielsen and Chuang's book. In section 4.2 they discuss that for any unit vector $\hat n$, the rotation operator $R_{\hat n}(\theta) = \exp(-i\theta\hat n \...
- 103
0
votes
1
answer
101
views
Show that any measurement where the measurement operators and the POVM elements coincide is a projective measurement
The following question is exercise 2.62 from Nielsen and Chuang's "Quantum Computation and Quantum Information"
Show that any measurement where the measurement operators and the POVM ...
- 295
1
vote
1
answer
54
views
Verify that if $A$ and $B$ are diagonal in the same orthonormal basis, then $[A,B]=0$
This is from Nielson and Chuang's textbook "Quantum Computation and Quantum Information".
They state the Simultaneous Diagonalisation Theorem: Suppose $A$ and $B$ are Hermitian operators. ...
- 295
0
votes
0
answers
57
views
Proof of the Lieb's theorem
Lemma A6.2: Let $R1 , R2 , S1 , S2 , T1, T2$ be positive operators such that
$0 = [R1, R2 ] = [S1, S 2 ] = [T1, T2 ]$, and
$$
R1 ≥ S1 + T1\\
R2 ≥ S2 + T2
$$
Then for all $0 ≤ t ≤ 1$,
$$
R_1^t R_2^{1−t}...
- 791
1
vote
0
answers
94
views
Why can Shor code fix arbitrary errors?
This is taken from Page 434 of Nielsen and Chuang:
To simplify the analysis, suppose noise of an arbitrary type is occurring on the first
qubit only; we’ll come back to what happens when noise is ...
- 147
1
vote
1
answer
44
views
Verification for calculation on Shor's code
Here I have tried to determine the end result for the qubit states, when we apply an arbitrary gate on the first qubit in the 9 qubit code.
I have followed this diagram:
U's operation on a qubit can ...
- 147
0
votes
1
answer
39
views
In quantum error correction, what does an "arbitrary error that yields an un-normalized state" mean?
This is from page 434 of Nielsen and Chuang:
. Supposing the state of the encoded qubit is |ψ⟩
before the noise acts, then after the noise has acted the state is
E(|ψ⟩⟨ψ|). To analyze the effects of
...
- 147
1
vote
1
answer
54
views
Non trace-preserving map in axiomatic approach to quantum operations
In Nielsen and Chuang's Quantum Computation and Quantum information there is an axiomatic definition of the quantum operation (as one of the 3 approaches to quantum operations).
A quantum operation is ...
- 157
0
votes
1
answer
80
views
Clarification regarding application of distributive property in "quantum teleportation" example
For context, this is from Page 27 of Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press:
She then sends the ...
- 147
1
vote
1
answer
59
views
Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?
In Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says:
A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i,
The ...
- 205
1
vote
2
answers
124
views
How to show that the three-qubit repetition code only corrects up to 1-bit flip errors?
From Nielsen and Chuang, the error correction criteria is $$P E_i^{\dagger} E_j P=\alpha_{i j} P$$ $P$ is the projector onto the correct codespace, $E_{j}$ are error operations and $\alpha_{i j} $ is ...
- 219
2
votes
2
answers
130
views
How are quantum error-correction conditions in Nielsen and Chuang implemented in practice?
Quantum error-correction conditions in Nielsen and Chuang, 10th-anniversary edition (Theorem 10.1) state that the error operation $\mathcal{E}$ with operation elements $\{E_i\}$ is correctable if and ...
- 157
1
vote
2
answers
199
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 ...
- 791
1
vote
1
answer
269
views
Nielsen Chang exercise 4.10
I' m trying to figure out how to decompose unitary operator U using only Rx and Ry rotations. I understand that I must use result of exercise 4.8 ($ U = e^{i\alpha}R_n(\theta)$). But I don't ...
- 13
1
vote
1
answer
68
views
Derivations of Equations (12.205 - 12.207) in Nielsen Chuang
While proving the security of the BB84 protocol, Nielsen and Chuang demonstrate that it is possible to reduce the CSS protocol to the secure BB84 protocol without requiring Alice to reveal the value ...
- 355
0
votes
1
answer
94
views
Do the parity matrices of $\text{CSS}(C_1, C_2)$ and $\text{CSS}_{u,v}(C_1, C_2)$ need to be same and the Number of Distinct Equivalent Descriptions?
This question relates to Exercise 10.27 of Nielsen & Chuang, which says:
Show that the codes defined by
$$| x + C_2 \rangle = \frac{1}{\sqrt{C_2}} \sum\limits_{y \in C_2} (-1)^{u \cdot y} | x+y+v ...
- 355
2
votes
2
answers
92
views
How Does the Transformation $|x \rangle | 0 \rangle \rightarrow | x \rangle | Hx \rangle $ Avoid Violating the No-Cloning Theorem?
This question relates to Nielsen & Chuang, Exercise 10.26, which says
Suppose $H$ is a parity check matrix. Explain how to compute the transformation $|x \rangle | 0 \rangle \rightarrow | x \...
- 355
1
vote
1
answer
39
views
Derivation of Equation 10.69 in Nielsen Chuang
In the proof of correctness of CSS codes in Nielsen and Chuang, we see that equation $(10.68)$:
$$
\frac{1}{\sqrt{|C_2|}} \sum\limits_{y \in C_2} (-1)^{(x+y)\cdot e_2} |x+y \rangle
$$
can be ...
- 355
1
vote
0
answers
47
views
What do we mean by family of CSS codes?
In proving the security of BB84 in Nielsen and Chuang (10th anniversary edition - Section 12.6.5), they argue that a codeword in $\text{CSS}(C_1, C_2)$ is represented by
$$\frac{1}{\sqrt{|C_2|}} \sum\...
- 355
1
vote
1
answer
127
views
Calculation of $\frac{d}{dt} I_t(A,X)$ in proving the convexity of the relative entropy via Lieb's theorem
In Page 520, Entropy and information, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that
The relative entropy $S(ρ||σ)$ is jointly convex in its arguments, where $S(ρ|...
- 791
2
votes
1
answer
98
views
How to extract probabilities from Kraus representation?
Consider a quantum operation described by Kraus operators $K_1, ..., K_n$. As I understand the effect of this operation on a density matrix $\rho$ can be described as $ \mathcal{E}(\rho)= \sum_{i}p(i)\...
- 157
0
votes
0
answers
53
views
The matrix norm $||A||=\max_{\langle u|u\rangle=1}|\langle u|A|u\rangle|$ in the proof of Lieb's theorem
In Exercise A6.4, Appendix 6: Proof of Lieb’s theorem, Page 645, Quantum Computation and Quantum Information by Nielsen and Chuang, A matrix norm of $A$ is defined as $$||A||=\max_{\langle u|u\rangle=...
- 791
4
votes
1
answer
105
views
How to correct error during the syndrome measurement or recovery process?
Here's the figure 10.21 from Nielsen's Quantum Computation and Quantum Information to explain the fault-tolerant quantum computing, where the circuit in the figure corrects the error that happens in ...
- 41
4
votes
0
answers
243
views
Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy
The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is
$$
S(A,B)\ge|S(A)-S(B)|
$$
this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
- 791
5
votes
3
answers
100
views
How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?
How can one prove that $tr(A\mid\psi\rangle\langle\psi\mid)=\langle\psi\mid A\mid\psi\rangle$? In Nielsen/Chuang they mention this is due to Gram-Schmidt decomposition but I can’t understand how.
2
votes
1
answer
107
views
What is the meaning of $\langle e_k|U|e_0\rangle$ when $U$ acts on a larger Hilbert space than that in which $|e_0\rangle$ and $|e_k\rangle$ live?
In Nielsen and Chuang, 10th Anniversary Edition, there is a definition of the operator sum representation of a quantum operation: $\mathcal{E}(\rho)=\sum_{k}\langle e_k|U[\rho\otimes|e_0\rangle\langle ...
- 157
5
votes
2
answers
208
views
Why is the operation in Nielsen and Chuang's Section 8.5 not a quantum operation?
In Section 8.5 of the 10th anniversary edition, Nielsen and Chuang discuss the limitations of the quantum operations framework. They give an example of a qubit prepared in an unknown state $\rho$, ...
- 171
3
votes
1
answer
105
views
Is the phase-estimation a specific case of the Hidden Subgroup Problem?
I read Nielsen & Chuang and I have difficulties understanding the links between the Hidden Subgroup Problem and the Phase Estimation.
In Exercise 5.14 (Section 5.3.1 "Application: order-...
- 81
0
votes
2
answers
72
views
Is there a criteria to ensure a one-qubit operator is exactly of the form $R_n(\theta)$ (i.e without a global phase $e^{i\alpha}$)?
Reading the Nielsen and Chuang, I saw that every unitary operator $U$ can be written as $e^{i\alpha} R_n(\theta)$ for some well chosen $n \in \mathbb{R}^3$ and $0 \leq \theta < 2\pi$.
I would like ...
- 81
3
votes
2
answers
532
views
How to choose a suitable number of iterations for Grover's algorithm?
In Nielsen and Chuang (2010), section 6.1.1. it is written:
"For an N item search problem with M solutions, it turns out that we need only apply the search oracle ...
- 51
2
votes
0
answers
74
views
Grover's algorithm for multiple solutions complexity
I'm reading Nielsen&Chuang book (for myself) and I'm completely stuck with one of the problems, 6.3(Database retrieval):
Given a quantum oracle which returns $\left|{k, y \bigoplus X(k)}\right>$...
1
vote
1
answer
59
views
Quick conditions to see that $I - E_1 -E_2$ is positive for $E_1, E_2$ positives (example from Nielsen and Chuang)
In the Nielsen and Chuang ("Quantum Computation and Quantum Information"), section 2.2.6 POVM measurements, they define these three operators:
$E_1 = \frac{\sqrt{2}}{1+\sqrt{2}} |1\rangle \...
- 81
2
votes
1
answer
222
views
Solution to Nielsen & Chuang Exercise 5.3 (FFT)
Can somebody help me with the solution of Nielsen Chuang, where we are supposed to derive the FFT from the equation (5.4):
$$|j_1,\ldots,j_n\rangle\rightarrow\frac{\big(|0\rangle+e^{2\pi i 0.j_n}|1\...
- 21
4
votes
1
answer
252
views
Question regarding Quantum Phase Estimation (Nielsen and Chuang exercise 5.8)
I was working through Nielsen and Chuang's book on quantum computing and they state the following result regarding the performance of the Quantum Phase Estimation algorithm,
"... given the input $...
2
votes
2
answers
186
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, ...
- 791
3
votes
0
answers
217
views
Prove $|η(r) − η(s)| ≤ η(|r − s|)$ when $|r − s| ≤ 1/2$ [closed]
Background
If $\rho$ and $\sigma$ are density matrices such that the trace distance between them satisfies $T(\rho,\sigma)\leq1/e$. Then the Fannes' inequality states that $$|S(\rho)-S(\sigma)|\leq T(...
- 791
1
vote
1
answer
196
views
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 ...
- 791