Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
2 answers
45 views

Why does it matter that Schmidt number is invariant under unitary transformations?

I am reading Nielsen & Chuang and they say this: "The bases $|i_A\rangle$ and $|i_B\rangle$ are called the Schmidt bases for A and B, respectively, and the number of non-zero values $\...
researcher101's user avatar
2 votes
3 answers
165 views

Why do we need/have the operator sum representation (Kraus representation)?

I am reading through Nielsen & Chuang, and I am on the section about operator sum representation. They performed this derivation. Why is it important and useful for us to bundle together the ...
researcher101's user avatar
1 vote
0 answers
31 views

Decomposing density matrix into arbitrary minimal ensemble

I came across this exercise in Nielsen & Chuang and am trying to understand it intuitively. Here is my reasoning of what is going on: The purpose of this exercise: Let’s say we are given a density ...
researcher101's user avatar
1 vote
2 answers
50 views

What is meant with "different ensembles can give rise to the same density matrix?"

I am reading the Nielsen & Chuang section on density matrices and I don't understand the example given to demonstrate a concept. Here is what I am reading: First, they said these two different ...
researcher101's user avatar
2 votes
0 answers
36 views

Reasoning behind unitary freedom in the ensemble for density matrices theorem

Although my question has the same title of a different question, it is not a duplicate. I am asking a different question. I don't care why it made it into the book. Here is a theorem from Nielsen &...
researcher101's user avatar
1 vote
2 answers
287 views

Why is a post-measurement state a valid quantum state?

Postulate 3 of the postulates of quantum mechanics in Nielsen & Chuang states that Quantum measurements are described by a collection ${M_m}$ of measurement operators... and the state of the ...
Josh's user avatar
  • 407
2 votes
1 answer
78 views

Quantum Error Correction Difficulties: "Measurements destroy quantum information" Intuition

Context: In section 10.1.1, Nielsen and Chuang describe the difficulties QEC faces compared to classical error correction. Particularly that Measurements destroy quantum information. Shor states in ...
vollautomatthi's user avatar
2 votes
0 answers
72 views

On the use of $\log(P\otimes Q)= \log P\otimes I+I\otimes\log Q$ for relations between entropic quantities. What if $P,Q$ are only semidefinite?

Many properties of entropic quantities are shown by resorting to related properties of the relative entropy of suitable quantities. For instance, subadditivity of entropy may follow from non ...
atlantropa's user avatar
2 votes
2 answers
144 views

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 ...
atlantropa's user avatar
2 votes
0 answers
62 views

Question about fault-tolerant measurement in Nielsen&Chuang page 490

The fault-tolerant procedures for state preparation and logical operation consider the procedure successful if only a one-qubit error occurs in each encoded block of the output. So, I thought that the ...
이호준's user avatar
2 votes
2 answers
206 views

Prove that $\text{Tr}(M|ψ\rangle\langleϕ|)=\langleϕ|M|ψ\rangle$

Question: I am studying alone, and I found p.76 of the book quantum computation and quantum information of nielsen &c huang that: $$\text{Tr}(M |\psi\rangle \langle\psi)=\langle\psi| M |\psi\...
OffHakhol's user avatar
  • 155
2 votes
1 answer
481 views

How to construct a quantum circuit for quantum Fourier transform in a prime dimensional Hilbert space?

This problem is given as a problem in Nielsen and Chuang. Consider a Hilbert space of dimension $p$ where $p$ is a prime number. Quantum Fourier transform (QFT) in this space is defined as $$ |j\...
Abu Saleh Musa's user avatar
2 votes
1 answer
228 views

Trace Distance in Bloch sphere, what is the vector of Pauli matrices?

While reading Chapter 9.2.1 Trace distance in "Quantum Computation and Quantum Information," I encountered a question. What is the vector of Pauli matrices referring to? $$ \vec{\sigma} = (\...
Wang Sheffield's user avatar
1 vote
1 answer
69 views

How to find projection operators for spectral decomposition

I am a little bit confused about the spectral decomposition for the observable $Z_{1}Z_{2}$ in Section $10.1$ of Nielsen and Chunag's "Quantum Computation and Quantum Information". The idea ...
am567's user avatar
  • 597
2 votes
0 answers
135 views

Solution Nielsen and Chuang exercize 10.71

Exercise 10.71: Verify that when $M = e^{−iπ/4}SX$ the procedure we have described gives a fault-tolerant method for measuring $M$. The book describes a procedure to perform the measurement. Instead ...
Maria 's user avatar
  • 21
0 votes
0 answers
27 views

Which stabilizer generators of a CSS code follow from which parity check matrices?

Do I have the right correspondence below between $X$-type ($Z$-type) stabilizer generators and the rows of the parity check matrices of $C_2^\perp$ ($C_1$)? I ask because it seems Nielsen and Chuang ...
user196574's user avatar
1 vote
2 answers
80 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 \...
qntdni's user avatar
  • 23
5 votes
1 answer
121 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 ...
DJD's user avatar
  • 53
2 votes
0 answers
115 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 ...
Sachindra Kumar's user avatar
2 votes
2 answers
105 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\...
karry's user avatar
  • 629
0 votes
2 answers
93 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|) ...
Matodo's user avatar
  • 67
1 vote
1 answer
309 views

Can a unital channel not be 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 ...
Sudhir Kumar's user avatar
4 votes
2 answers
443 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) ...
Matodo's user avatar
  • 67
1 vote
3 answers
139 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{...
Matodo's user avatar
  • 67
1 vote
1 answer
66 views

Why is the matrix obtained from the coefficients of orthogonal states unitary?

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
1 vote
1 answer
102 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. ...
Physkid's user avatar
  • 518
4 votes
2 answers
179 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 ...
Marcus's user avatar
  • 41
2 votes
0 answers
96 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/...
Jintao Yu's user avatar
0 votes
0 answers
145 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 ...
am567's user avatar
  • 597
5 votes
2 answers
776 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 \...
slimmerikko's user avatar
0 votes
1 answer
163 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 ...
am567's user avatar
  • 597
1 vote
1 answer
62 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. ...
am567's user avatar
  • 597
0 votes
0 answers
71 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}...
Sooraj S's user avatar
  • 831
1 vote
0 answers
136 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 ...
Alan Whitteaker's user avatar
1 vote
1 answer
58 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 ...
Alan Whitteaker's user avatar
0 votes
1 answer
70 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 ...
Alan Whitteaker's user avatar
1 vote
1 answer
58 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 ...
EugeneB's user avatar
  • 167
0 votes
1 answer
81 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 ...
Alan Whitteaker's user avatar
1 vote
1 answer
67 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 ...
Guangliang's user avatar
1 vote
2 answers
219 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 ...
Aubrey Sharansky's user avatar
2 votes
2 answers
196 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 ...
EugeneB's user avatar
  • 167
2 votes
2 answers
319 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 ...
Sooraj S's user avatar
  • 831
1 vote
1 answer
397 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 ...
Samir Akhmed's user avatar
1 vote
1 answer
71 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 ...
Josh's user avatar
  • 407
0 votes
1 answer
131 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 ...
Josh's user avatar
  • 407
2 votes
2 answers
104 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 \...
Josh's user avatar
  • 407
1 vote
1 answer
49 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 ...
Josh's user avatar
  • 407
1 vote
0 answers
56 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\...
Josh's user avatar
  • 407
1 vote
1 answer
135 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|...
Sooraj S's user avatar
  • 831
3 votes
1 answer
128 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)\...
EugeneB's user avatar
  • 167

1
2 3 4 5
7