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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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
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2 answers
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Performing a projective measurement, is the resulting expectation value $\langle \Psi|M|\Psi\rangle$ bounded between $+1$ and $-1$?

Suppose we have a quantum state $|\Psi\rangle = \alpha|0\rangle + \beta|1\rangle$.According to a measurement operator M, the projective measurement of $|\Psi\rangle$ is given by $\langle\Psi|M|\Psi\...
aghin00's user avatar
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-2 votes
1 answer
51 views

help understanding gate to hamiltonian and representation

So I have this question: Given an operator, find some Hamiltonian implementing this operator/gate. I have realized that this is a swap gate and I know the matrix for it. I also know that $U = \text{...
George's user avatar
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1 vote
1 answer
45 views

conditions for two hermitians operators same up to unitary

Let $A$ and $B$ $2^n \times 2^n$ Hermitian matrices. What are sufficient and necessary conditions that they are equal up to some unitary, i.e. there exists $U$ such that $A = U B U^\dagger$? The first ...
Jon Megan's user avatar
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0 answers
33 views

How to take partial trace of a $n - 1$ qubit subsystem from a $n$ qubit system

I would like to calculate the expression $$ \text{Tr}_2\left\{R^z \sigma\right\}\,,$$ where $$\sigma = \rho \otimes |0\rangle \langle0|^{{\otimes}n}\,.$$ Here $$R = \sum{\theta_m}G_m\,,$$ $$\theta_m \...
Sudhir Kumar's user avatar
1 vote
1 answer
73 views

Quantum teleportation of unknown qubit when the entangled state is not a Bell state

Assume Bob and Alice have two particles with a prior entanglement: $A$ and $B$. The entangled state $|Ψ⟩$ is maximally entangled, and $$|Ψ⟩ = \frac{1}{\sqrt{2}}(|00⟩ + j|11⟩)\,,$$ where $j$ is a ...
Dogukan's user avatar
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2 votes
1 answer
168 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
0 votes
1 answer
53 views

How does a three-qubit state evolve through a CNOT gate?

Suppose I have a qubit which is entangled with another; let's say they are in $A|00\rangle+B|11\rangle$. If I have another qubit in the state $a|0\rangle+b|1\rangle$ then the combined state is $Aa|000\...
Tamilaruvi Saravanan's user avatar
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1 answer
40 views

Understanding the error operator representation $E = i^{\lambda}X(a)Z(b)$

Question regarding exercise $27.3.2$ in "Concise Encyclopedia of Coding Theory". The exercise states: We write $E = X((0,1))Z((0,0))$ and $E' = iX((0,1))Z((1,1))$. We choose the ordering $(...
am567's user avatar
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2 votes
0 answers
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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 ...
Maria 's user avatar
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2 votes
1 answer
181 views

How to find the eigenvectors and eigenvalues of a hermitian operator?

While reading Theoretical Minimum by Leonard Susskind, I came across the exercise 3.4 where he asked to find the eigenvalues and the eigenvectors of the matrix that represents the $\sigma_{n}$ ...
zizaaooo's user avatar
2 votes
1 answer
71 views

Error 'LocalSimulator' with Googlecolab

I have an error when I want to run the 'LocalSimulator'. I am not inside AWS, its mean I runnig from Google Colab. The code is the same on the notebooks from ...
Sandra Rosales's user avatar
0 votes
1 answer
59 views

What does "the eigenvectors of a Hermitian operator are a complete set" mean?

I read in my book that the eigenvectors of a Hermitian operator are a complete set. What does the author mean by that?
zizaaooo's user avatar
1 vote
0 answers
64 views

Distinguish two states with their priors probability

EDIT: This is a computer programming / coding exercise The states $\left|\psi\right>$ and $\left|\phi\right>$ are defined as $∣ϕ⟩=\cos(θ_ϕ)\left|0\right>+\sin(θ_ϕ) \left|1\right>$ with ...
Minh Triet's user avatar
-2 votes
1 answer
70 views

Measuring probabilities of 0 or 1 in a two qubit state

I'm preparing for my upcoming exam, I need to determine the probabilities with which Bob measures 0 or 1 and in both cases describe the state of Alice, this is my state: $$ |\psi\rangle =\frac{1}{\...
embe99's user avatar
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1 vote
1 answer
116 views

Is a linear combination of unitaries unitary?

Suppose you have a pure state $\vert\psi\rangle$. Consider the following operation. For unitaries $U_1$ and $U_2$, one can take complex numbers $\alpha, \beta$ where $|\alpha|^2 + |\beta|^2 = 1$ and ...
user1936752's user avatar
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0 votes
1 answer
44 views

Calculate of theoretical probabilities for the outcomes

I have a $|+\rangle$ state qubit and I measure it in a random basis. The random basis is made with random $\theta$, $\varphi$ and $\lambda$ of $U3$ gate. How can I calculate the theoretical ...
hdsa's user avatar
  • 9
2 votes
0 answers
51 views

Measuring an entangled quantum state

I have this exercise to solve, but I can't figure out how to proceed. First, I don't think the state proposed is valid (the probabilities don't sum up to 1). 'Secondly, given that it should be an ...
Giulia's user avatar
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Deriving a equation in "Data re-uploading for a universal quantum classifier" paper: U(phi1,phi2,phi3) U(x1,x2,x3) = U(theta + w *x)

In "Data re-uploading for a universal quantum classifier" paper the U(phi1,phi2,phi3) U(x1,x2,x3) is derived as U(theta + w *x) in compact state. How to derive the above equation ?
Aakash's user avatar
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1 vote
1 answer
27 views

Is qiskit.pulse.SamplePulse deprecated?

I'm currently following along the book "Learn Quantum Computing with Python and IBM Quantum Experience" by Rodert Loredo and I am at chapter 8 - Generating pulse schedules on hardware. The ...
Johannes Jyrgenson's user avatar
5 votes
2 answers
128 views

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 ...
junghyunHa's user avatar
0 votes
1 answer
81 views

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.
user27383's user avatar
5 votes
1 answer
117 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
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3 votes
1 answer
92 views

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 ...
metaUser's user avatar
0 votes
2 answers
50 views

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 $\hat{n}$. ...
Physkid's user avatar
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1 vote
0 answers
60 views

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 ...
Physkid's user avatar
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3 votes
2 answers
111 views

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 ...
YaGoi Root's user avatar
0 votes
1 answer
111 views

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.
shiranrubatsirorwashe's user avatar
1 vote
0 answers
31 views

Why does unitary matrix acts only on input qubit state of a vector that is a result of add modulo 2?

Let $|\psi\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k \oplus b\rangle $ so that $|\psi'\rangle = \frac{1}{\sqrt{2}}[\sum_{k=2}^{1}(-1)^{ka}(U_{A}|k\rangle \otimes U_{B}|k \oplus b\rangle)]...
Physkid's user avatar
  • 520
2 votes
1 answer
106 views

What is the expression for $|\psi\rangle\!\langle\psi|$ if $|\psi\rangle=\cos(\theta/2)|0\rangle+\sin(\theta/2)e^{i\phi}|1\rangle$?

Let $|\psi\rangle = \alpha|0\rangle + \beta |1\rangle$. In Bloch sphere representation, this is $\cos\frac{\theta}{2}|0\rangle + \sin\frac{\theta}{2}e^{i\phi}|1\rangle$. In matrix representation: $|\...
Physkid's user avatar
  • 520
1 vote
1 answer
80 views

Why can $(0,0,3/5,0,0,0,4/5,0,0)$ be written as $\frac35|3\rangle+\frac45|7\rangle$?

Context. $\newcommand{\qr}[1]{\left|#1\right\rangle}$ 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 ...
user1145880's user avatar
2 votes
2 answers
75 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
  • 679
2 votes
1 answer
77 views

Does the state obtained flipping $a,b$ in the state $(a,b)^T$ have a name?

Suppose we have a qubit with a state vector of $\begin{pmatrix} a \\ b \end{pmatrix} $. If we flip $a$ with $b$ does the new qubit has a name in relation to the first qubit?
Cerise's user avatar
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-1 votes
2 answers
38 views

Define a traceless part of $\rho$ [closed]

I saw in a paper: $|\bar{\rho}\rangle\rangle=|\rho\rangle\rangle-|\hat{I}\rangle\rangle / 2^{n / 2}$ for the $4^n$-dimensional vector representing the traceless part of $\rho$. https://arxiv.org/abs/...
karry's user avatar
  • 679
3 votes
1 answer
97 views

calculate the reduced density matrix of a 2 qubit state and compare the eigenvalues

So I have the exercise to apply a Cz gate to the following 2 Qubit state $|a\rangle \otimes |b\rangle = (a_0 |0\rangle + a_1 |1\rangle) \otimes (b_0 |0\rangle + b_1 |1\rangle)\\\\$ Afterwards, I ...
Ruebli's user avatar
  • 31
1 vote
0 answers
137 views

Do solutions to the exercises in Preskill's lecture notes exist anywhere?

I'm teaching myself quantum computing and would like to know if solutions to the exercises in Preskill's lecture notes exist anywhere. It's quite hard to see if my approach is correct because I don't ...
requiemman's user avatar
1 vote
0 answers
43 views

How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = Tr_{2}[\rho_{12}] = \sum_{i}\langle i_{2} | \rho | i_{2} \rangle$?

Let the system be a 2 - qubit system and let $\rho_{12}$ be a density matrix of some state for this 2 - qubit system. How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = Tr_{2}[\...
Physkid's user avatar
  • 520
2 votes
1 answer
109 views

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 ...
Ri dev's user avatar
  • 21
0 votes
2 answers
85 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
0 votes
1 answer
194 views

Unital channel which is not 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
1 answer
95 views

How do I find the reduced density matrix of a system where two people share one qubit and have one qubit of their own?

I have the following problem and have attempted to find a solution to it, but to no avail. Alice and Bob have one qubit each, say $|\psi\rangle$ with Alice and $|\phi\rangle$ with Bob. They also share ...
requiemman's user avatar
1 vote
3 answers
131 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
5 votes
2 answers
926 views

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 ...
X0-user-0X's user avatar
2 votes
0 answers
39 views

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 ...
Physkid's user avatar
  • 520
2 votes
5 answers
353 views

How to eliminate the global phase of a state vector?

Say that I have a qubit that began in the $|0\rangle$ state and then the Hadamard gate is applied, resulting in the following state: $ \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{...
bddicken's user avatar
  • 143
0 votes
0 answers
58 views

What is the value of $p(+)$?

I know the formula is $p = \left<\psi\right|M_{m}^{\dagger} M_{m}\left|\psi\right>$, where $\left|\psi\right>= \alpha\left|0\right>+\beta\left|1\right>$ and $M_m=\left|+\right> \left&...
karael's user avatar
  • 1
1 vote
1 answer
96 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
  • 520
2 votes
1 answer
447 views

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 ...
zizaaooo's user avatar
2 votes
0 answers
73 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
1 vote
1 answer
54 views

What is the probability of a state $|0\rangle$ being in another state $\alpha|0\rangle+\beta|1\rangle$?

I am trying to calculate the probability of a state (density matrix) being in a specific other state. Lets say I have a 2-dimensional state with the states given by the orthonormal basis states $|0\...
TTa's user avatar
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