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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Mathematical properties used to derive Kraus operators
In this answer, it was very well explained why Kraus operators are not numbers as it might seem when reading Nielsen and Chuang for the first time. I have a minor, purely technical and probably simple ...
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What is the use of the given circuit identity?
In Exercise 4.20 in Nielsen & Chuang we are asked to prove the circuit identity $(H_1 \otimes H_2) CX_{1,2}(H_1 \otimes H_2) = CX_{2,1}$
which is done technically by definition.
As a followup, it ...
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Quantum Measurement Postulate's Physical Meaning
I have always struggled to appreciate the measurement postulate (Section 2.2.3 of Nielsen and Chuang).
"Quantum Measurements are described by a collection of $\{M_m\}$ of measurement operators&...
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Exercise 7.40 in Nielsen & Chuang
I am currently going through Nielsen Chaung and having a go at all exercises but I am not sure how to treat this question in particular.
Exercise 7.40: (Refocusing dipolar interactions) Give a ...
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Fallacy of special significance of eigenvalues and eigenvectors of density operator
This question is an addition to the following question.
Nielsen and Chuang open the discussion of the unitary freedom on the ensemble for density matrices by pointing out the common fallacy to suppose ...
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Polar decomposition of $\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}$
By polar decomposition of a square invertible matrix $A$, I understand $A = |A| U$ for some unitary matrix $U$, where $|A| = \sqrt{A^\dagger A}$ with $\dagger$ denoting the conjugate-transpose ...
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In exercise 8.23 of Nielsen and Chuang why is the quantum operation no longer trace-preserving?
My question is about this exercise from Nielsen & Chuang:
For context, $\mathcal{E}_{AD}$ is the quantum operation for amplitude damping on a single physical qubit, with operation elements given ...
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What is the origin of the term POVM?
In Nielsen Chuang section 2.2.6 as they are introducing POVMs they write “The acronym POVM stands for ‘Positive Operator-Valued Measure’, a technical term whose historical origins we won’t worry about....
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Single-qubit quantum channel from the CNOT gate
I am studying quantum noise, chapter $8$ in Nielsen and Chuang. Section $8.2.2$ introduces an example for the definition of quantum operations, in particular the CX gate is introduced as an example. I ...
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No-cloning theorem and distinguishing between two non-orthogonal quantum states revisited revisited
There are many posts to this question from Nielson and Chuang's Quantum Computation and Quantum Information Exercise 1.2 page 57. It is required to prove that if a hypothetical device exists, which ...
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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 $\...
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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 ...
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Why do minimal ensemble decompositions for $\rho$ contain $|\psi⟩\in{\rm supp}(\rho)$ with probability $1/\langle\psi|\rho^{-1}|\psi⟩?$
I came across the following exercise (2.73) 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 ...
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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 ...
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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 &...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: $$\text{Tr}(M |\psi\rangle \langle\psi)=\langle\psi| M |\psi\...
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How to construct a quantum circuit for quantum Fourier transform in a prime dimensional Hilbert space?
This problem is given as a problem in Nielsen and Chuang. Consider a Hilbert space of dimension $p$ where $p$ is a prime number. Quantum Fourier transform (QFT) in this space is defined as
$$
|j\...
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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?
$$
\vec{\sigma} = (\...
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How to find projection operators for spectral decomposition
I am a little bit confused about the spectral decomposition for the observable $Z_{1}Z_{2}$ in Section $10.1$ of Nielsen and Chunag's "Quantum Computation and Quantum Information".
The idea ...
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Solution Nielsen and Chuang exercize 10.71
Exercise 10.71: Verify that when $M = e^{−iπ/4}SX$ the procedure we
have described gives a fault-tolerant method for measuring $M$.
The book describes a procedure to perform the measurement. Instead ...
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Which stabilizer generators of a CSS code follow from which parity check matrices?
Do I have the right correspondence below between $X$-type ($Z$-type) stabilizer generators and the rows of the parity check matrices of $C_2^\perp$ ($C_1$)? I ask because it seems Nielsen and Chuang ...
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Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
I have trouble understanding a proof in Nielsen & Chuang, specifically the identity in (10.20), which reads $$ U_k^\dagger P_k F_l \sqrt{\rho} = U_k^\dagger P_k^\dagger F_l P \sqrt{\rho}.$$
By ...
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Question about Nielson & Chuang Problem 9.2
I am working on the following problem from the book "Quantum Computation and Quantum Information" by Nielsen and Chuang.
Problem 9.2: Let $\mathcal{E}$ be a trace-preserving quantum
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Quantum Process Tomography for 2 qubits
I need clarification on a few aspects related to Box 8.5 and Exercise 8.34 from the book Quantum Computation and Quantum Information by Nielsen & Chuang .
While attempting Exercise 8.34, I ...
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What we get when measure $|0\rangle$ under computational basis?
It is said if we have been given the state $|0\rangle$, the measurement will yield $0$ with probability $1$ in Nielsen's book.
So here, the measurement will yield $0$ refers to we will get state $|0\...
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Explanation of the 2.60 equation page 76 in the Nielsen and Chuang [duplicate]
In the Nielsen and Chuang book page 76, equation 2.60 says that we can rewrite the trace $$Tr(A \left|\psi\right>\left<\psi\right|)$$ as follow :
$$Tr(A \left|\psi\right>\left<\psi\right|) ...
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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 ...
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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{...
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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{|...
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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.
...
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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 ...
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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/...
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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 ...
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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 \...
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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 ...
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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. ...
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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 ...
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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 ...
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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
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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