# 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 ...
1 vote
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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 ...
1 vote
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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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### 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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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/... 0 votes 0 answers 156 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 ... • 705 6 votes 2 answers 1k 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 \... • 113 0 votes 1 answer 193 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 ... • 705 1 vote 1 answer 65 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. ... • 705 1 vote 0 answers 148 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 ... 1 vote 1 answer 64 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 ... 0 votes 1 answer 79 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 ... 1 vote 1 answer 65 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 ... • 197 0 votes 1 answer 85 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 ... 1 vote 1 answer 68 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 ... • 237 2 votes 2 answers 257 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 ...
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 ...