# Questions tagged [textbook-and-exercises]

Applies to questions of primarily educational value - styled in the format similar to that found in textbook exercises. This tag should be applied to questions that are (1) stated in the form of an exercise and (2) at the level of basic quantum information textbooks.

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### Decomposition of a $4 \times 4$ unitary matrix

I am currently studying the paper "Decomposition of unitary matrices and quantum gates (2012)" and referring to the textbook Quantum Computation and Quantum Information. Among the topics, I ...
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### How to show that the GHZ state is absolutely maximally entangled?

A multipartite state is called absolutely maximally entangled if for its any bipartition the reduced density matrix of smaller part is maximally mixed. Show that GHZ state has this property.
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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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### Why is the operator $M_a |x\rangle= |a \cdot x \pmod{N} \rangle$ unitary?

If $N\geq 2$, $a\in \mathbb{Z}_N$, and $a^r= 1$ for some $r$. Consider the operator $M_a$, which is related to order finding : $M_a |x\rangle= |a \cdot x \pmod{N} \rangle$ if $x\in \mathbb{Z}_N$ What ...
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### Why is a density matrix an orthogonal projector?

Suppose I have a density matrix like $\rho = \frac{1}{2}[I + \hat{n}\vec{\sigma}]$. The claim is that $\rho$ is an orthogonal projector for the state $|+\rangle$ in an arbitrary direction $\hat{n}$. ...
1 vote
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### Possible post - measurement states for Bell state $\frac{1}{\sqrt{2}}[|00\rangle + |11\rangle]$

This is in reference to page 241 of Introduction to classical and quantum computing by Thomas.G Wong. The author starts off with a Bell state $\frac{1}{\sqrt{2}}[|00\rangle + |11\rangle]$. In trying ...
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### How to know what eigenvalue corresponds to measurements of individual qubits in a multiqubit system?

I'm working through the book "Introduction to the Theory of Quantum Information Processing" by Bergou and Hillary, and I've encountered a scenario that I'm not sure how to approach. In ...
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### The expectation values for the values of both qubits [closed]

Let’s consider the two-qubit state |Ψ⟩ =(1/2)|00⟩ + i(√3/4)|01⟩ +(3/4)|10⟩. a) Find the expectation values for the values of both qubits separately.
1 vote
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1 vote
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### Why can $(0,0,3/5,0,0,0,4/5,0,0)$ be written as $\frac35|3\rangle+\frac45|7\rangle$?

Context. $\newcommand{\qr}{\left|#1\right\rangle}$ A passage from a lecture by Scott Aaronson: "As an example, instead of writing out a vector like $$(0,0,3/5,0,0,0,4/5,0,0),$$ you can simply ...
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### How to figure out whether a truth table can correspond to a valid quantum gate

I am new to quantum computing and trying to wrap my head around this exercise from Wong's introduction to classical and quantum computer. I can interpret it mentally that first is a valid quantum ...
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### What are "completely positive" and "CPTP" quantum maps?

I am studying quantum computing a little bit by myself, and I have simple questions. I didn't find a clear definition of what is a completely positive and trace-preserving (CPTP) map. The best I've ...
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### Mechanics of expanding projector operator (two - qubits) in basis of traceless Hermitian Paul operators

I am currently on a set of lecture notes which says that for a state vector $| \psi \rangle_{AB}$ describing a tensor product state, its density operator $| \psi \rangle \langle \psi |_{AB}$ can be ...
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1 vote
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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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### Prove that the eigenvectors of a Hermitian operator form a basis

While I was reading the book Quantum Mechanics The Theoretical Minimum, the author said that if a vector space is $N$ dimensional, an orthonormal basis of $N$ vectors can be constructed from ...
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### How to show that the trace distance equals the maximal total variation distance?

Let $\rho$ and $\sigma$ be two density operators such that probability of obtaining $a$ is $tr(\rho E_a)$ if the state before measurement was $\rho$ and $tr(\sigma E_a)$ if the state before ...
1 vote
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I'm following a lesson, and it says that the Hadamard gate can be decomposed to three gates: RZ(pi/2), squared root Z, and RZ(pi/2). However, when I do matrix multiplication of these three matrices, I ...
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### Derivation explanation needed [closed]

I'm pretty new to the field. I was reading Preskill Ph219 course notes and came across this. I am a bit confused about the derivation and wondered if someone can write down some skipped steps here.
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### Is the quantum state fidelity defined as $|\langle\psi|\phi\rangle|$ or $|\langle\psi|\phi\rangle|^2$? [duplicate]

I am reading Preskill's notes on quantum information as well as Chuang's textbook. I saw that fidelity is defined in two different ways in Preskill's notes and Chuang's book. In Preskill's notes, ...
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### Link between Lagrange Multipliers in Quantum Optimal Control and Traditional Method

In the context of quantum optimal control, the term "Lagrange multiplier" has been used, which seems to draw a connection to the traditional Lagrange multiplier method used in problems of ...
1 vote
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1 vote
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### Why does Sakurai's book use the gyromagnetic ratio of the electron as $\frac{e}{m_e c}$ instead of $\frac{e}{m_e}$?

I'm studying quantum computing, but I had to review some concepts in supplementary readings, where I came across the following question about a simple detail: Why does Sakurai's book (near Eq. 1.1) ...
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### Definition of fault-tolerant measurement in QCQI

In the book "Quantum Computation and Quantum Information," one of the definitions for fault-tolerant measurement is mentioned as "any single component in the procedure results in an ...
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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 ...
1 vote
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### Can we rearrange terms in the tensor product?

Define $o = A \otimes B$. Compute the results of $o^{\otimes N}$. We have \begin{align} o^{\otimes N} &= (A \otimes B) \otimes (A \otimes B) \otimes ... \otimes (A \otimes B). \end{align} Can we ...
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### Unitary matrix knowing input and ouput states

Input: n qubits after H gates. Output, after measure: the n different bit strings of length 2^n with just one 1, with same probabilities. Or said differently: Input state |phi0>=(1,1,...1)/2^n ...
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### Can you distinguish between $|0\rangle, |1\rangle$, and $\frac{1}{\sqrt 2} (|0\rangle + |1\rangle)$?

(A beginner here; possibly a stupid question. Please be gentle. Sorry if I used a wrong tag.) Suppose that I receive a (classically) random number, which is either $1$ or $2$ or $3$. Depending on this ...
Suppose the below operator  x\sum_{n=0}^{\infty}\tanh^{2n}(r)|0,n\rangle\langle 0,n| +y\sum_{n=0}^{\infty}\tanh^{2n}(r)|1,n\rangle\langle 1,n| +z\sum_{n=0}^{\infty}\tanh^{2n}(r)(n+1)|1,n\rangle\...
### How would you draw the phase-estimation circuit for the eigenvalues of $U = \mathrm{diag}(1,1,\exp(\pi i/4),\exp(\pi i/8))$?
How would you draw the phase-estimation circuit for the eigenvalues of: $U = \mathrm{diag}(1,1,e^{(\pi i)/ 4}, e^{(\pi i)/8})$ corresponding to the eigenstates $|10\rangle$ and $|11\rangle$? What is ...