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.

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Are unital channels always mixed-unitary?

How to prove mixed unitary of a channel for a multi-qubit system is not Unital. I am trying to prove Problem 8.3 of Nelson and Chuang's book. Here's a snippet of the text: Shall I need to take two ...
Sudhir Kumar's user avatar
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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) ...
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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{...
Matodo's user avatar
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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{|...
orangonabbo's user avatar
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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. ...
Physkid's user avatar
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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 ...
Marcus's user avatar
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$T_1$ 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/...
Jintao Yu's user avatar
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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 ...
Aislin_367's user avatar
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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 \...
slimmerikko's user avatar
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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 ...
Aislin_367's user avatar
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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. ...
Aislin_367's user avatar
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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}...
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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 ...
Alan Whitteaker's user avatar
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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 ...
Alan Whitteaker's user avatar
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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 ...
Alan Whitteaker's user avatar
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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 ...
EugeneB's user avatar
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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 ...
Alan Whitteaker's user avatar
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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 ...
Guangliang's user avatar
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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 ...
Aubrey Sharansky's user avatar
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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 ...
EugeneB's user avatar
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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 ...
Sooraj S's user avatar
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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 ...
Samir Akhmed's user avatar
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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 ...
Josh's user avatar
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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 ...
Josh's user avatar
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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 \...
Josh's user avatar
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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 ...
Josh's user avatar
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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\...
Josh's user avatar
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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|...
Sooraj S's user avatar
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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)\...
EugeneB's user avatar
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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=...
Sooraj S's user avatar
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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 ...
SUSY's user avatar
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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 ...
Sooraj S's user avatar
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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.
Alberto Zorzato's user avatar
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1 answer
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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 ...
EugeneB's user avatar
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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 $\rho$, ...
aquohn's user avatar
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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-...
user8622655's user avatar
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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 ...
user8622655's user avatar
2 votes
2 answers
345 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 ...
Lucas's user avatar
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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 $\left|{k, y \bigoplus X(k)}\right>$...
Михаил Горчаков's user avatar
1 vote
1 answer
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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 \...
user8622655's user avatar
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1 answer
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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\...
Gregor123's user avatar
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1 answer
190 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 $...
Bikrant Bhattacharyya's user avatar
2 votes
2 answers
174 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, ...
Sooraj S's user avatar
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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(...
Sooraj S's user avatar
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1 answer
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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 ...
Sooraj S's user avatar
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1 answer
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clarifying a step in the proof of Solovay-Kitaev theorem

There is a step in the proof of the proof of Solovay-Kitaev theorem about the existence of a set containing words of at most length length $l_0$ that cover $SU(2)$ . The proof I'm reading in given in ...
madeel's user avatar
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What is the definition of expectation value in Nielsen and Chuang?

As someone who learned Quantum Mechanics prior to studying Quantum Computing, I am having problems with this book. For example, equation 2.92 is incorrect. That is the definition of expected value, ...
Thomas Rigby's user avatar
2 votes
1 answer
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How is the $\beta$-matrix interpreted in single qubit QPT?

In Chapter 8 of Quantum Computation & Quantum Information by Nielsen & Chuang, more precisely Box 8.5, there is an example of quantum process tomography for a single qubit. (The same ...
Calle's user avatar
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Proof of Nielsen's theorem (Theorem 12.15) given in Nielsen-Chuang (assumption of invertibility)

Theorem 12.15 of Nielsen and Chuang's 10th anniversary edition is Nielsen's Theorem (1999). In particular, it says, Theorem 12.15: A bipartite pure state $\mid \psi \rangle$ may be transformed to ...
MathFrak96's user avatar
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1 answer
176 views

In what sense is $\langle\psi|\rho|\psi\rangle$ the overlap between $|\psi\rangle$ and $\rho$?

The fidelity between a pure state $|\psi\rangle$ and an arbitrary mixed state $\rho$ is given by, $F(|\psi\rangle,\rho)=\sqrt{\langle\psi|\rho|\psi\rangle}$, which is stated to be equal to the square ...
Sooraj S's user avatar
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