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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1answer
38 views

How do I apply a matrix to a ket state?

If we have the following matrix: $$\frac{1}{\sqrt{2}}\begin{pmatrix}1&1&0&0\\ 1&-1&0&0\\ 0&0&1&-1\\ 0&0&1&1\end{pmatrix}$$ How do we find the output for ...
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1answer
31 views

Knill-Laflamme condition derivation in Nielsen&Chuang: issue to understand a part of the proof

I have some trouble to understand the proof in Nielsen&Chuang about Knill-Laflamme conditions. The conditions: Let $C$ be a quantum code and $P$ the projector onto $C$. Suppose $\mathcal{E}$ is a ...
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2answers
74 views

Show when $a_k$ and $b_k$ are correlated when measuring in different bases, in the BB84 protocol

I'm trying to answer the following question about the BB84 protocol from Nielsen and Chuang's Introduction to Quantum Information. As I understand it, the string $b$ is determining whether we are ...
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2answers
69 views

In quantum process tomography, how does $\chi$ characterize a quantum process?

I'm working through Nielsen and Chuang and I'm pretty confused by the discussion of quantum process tomography. I'm trying to work through an example of 1-qubit state tomography given by N&C (box ...
3
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1answer
121 views

What is the quantum Fourier transform of $\alpha|0\rangle+\beta|1\rangle$?

Given $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$ and $|\alpha|^2 + |\beta|^2 = 1$, what would the quantum Fourier transform of $|\psi\rangle$ be? I know it is of the form $\frac{1}{\sqrt{2}}(...
3
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1answer
65 views

How do I show that $R_z(\theta)=e^{-iZ\theta/2}$?

I know that an $R_z (\theta)$ gate is equivalent to the unitary transformation $e^{-iZ * \theta/2}$ but I'm not sure how we get there. I know that for every Hermitian matrix there is a corresponding ...
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2answers
74 views

Is the tensor product of 2 Hadamard gates entangled?

Assume that you have a system of two qubits in the state $|11 \rangle$. Apply $H \otimes H$, where $H$ is the Hadamard matrix. Is the state $(H \otimes H)|11\rangle$ entangled? I know if we take the ...
2
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1answer
71 views

Given $|\psi\rangle=(U_A\otimes U_B)|0,0\rangle$, is $|\psi\rangle\!\langle\psi|$ always a product state?

say I have some state in the combined space $\psi$ ∈ $H_A\otimes H_B$, where $\psi=U_A \otimes U_B|0,0\rangle$ (operators from respective spaces), and $\rho_A, \rho_B$ the respective density matrices. ...
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1answer
57 views

How to perform a projective measurement on one component of a composite system?

For simplicity, let $|\phi\rangle|\psi\rangle\in\Bbb C^2\otimes\Bbb C^2$. I know how to compute the projective measurement $\{P_m\}_m$ of $|\phi\rangle|\psi\rangle$ on $\Bbb C^2\otimes\Bbb C^2$, but I ...
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2answers
56 views

How to prove that EPR outcomes have equal probability no matter the basis?

Recently in class, we learned about the EPR state. I know that no matter what basis the first qubit is measured in, the two outcomes have an equal probability. However, how does one prove this? I ...
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2answers
68 views

Exercise 4.6 in Quantum Computing and Quantum Information Nielsen and Chuang

Question 4.6: One reason why the $R_\hat{n}(θ)$ operators are referred to as rotation operators is the following fact, which you are to prove. Suppose a single qubit has a state represented by the ...
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1answer
73 views

PyTorch and Qiskit example from the Qiskit textbook seems broken

After executing the proceeding code blocks, when I try to copy the same code from Qiskit textbook on my jupyter notebook, I get the error as ...
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3answers
151 views

Why does the trace of density operators need to be one?

Usually, the textbook starts with a few assumptions of what density operator $\rho$ has. One of them is $Tr(\rho) = 1$. Why is that?
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3answers
320 views

Can we write Pauli-Y gate without even complex part?

I was just curious, why is the quantum gate Y-gate (Pauli-Y gate) written in terms of complex numbers? We can actually write Pauli-Y gate as $$ Y = i * \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{...
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2answers
91 views

Are these two 'divided by two' terms related?

I have a question about the two equations: Any matrix in $SU(2)$ could be parametrized as $$ R_{\hat{n}}(\theta) = \cos\left(\frac{\theta}{2}\right)I-i\sin\left(\frac{\theta}{2}\right)(\hat{n}\cdot\...
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2answers
68 views

Show that the Bell states form a basis

I can't seem to understand how to show that the Bell states for a basis. Should I explain that through the circuit and what gates are used or by the basic proof behind proving a set as a basis?
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0answers
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Question regarding bloch vector solutions for master equation on page 388 of N&C

On page 388 of N&C, you are asked to find the solution to a differential equation for a two-level atom coupled to a vacuum. However, I have no experience with differential equations, so I am ...
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1answer
40 views

How can I factor Hadamard gates according to $U=e^{i\delta}R_x(\alpha)R_y(\beta)R_z(\gamma)$

According to the formula $U=e^{i\delta}R_x(\alpha)R_y(\beta)R_z(\gamma)$, We know that a single quantum gate can be decomposed arbitrarily, But according to the book Quantum Computation. by Nielsen, I ...
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3answers
170 views

Is there any difference between “value of a qubit” and its “state”?

Value of a qubit and its state - is there any difference between these two terms in sense of terminology? For example, can we name this state of a qubit also a value of a qubit: $$ |\psi\rangle = \...
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1answer
46 views

How to calculate the overlap of the orthogonal state?

This is probably a very obvious question, but I am going through this problem set and I don't understand why in 1b) it says that it is obvious that $|\langle\psi_1^\perp|\psi_2\rangle|=\sin\theta$ ...
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2answers
76 views

Worked example of Bernstein-Vazirani - understanding bitwise product

From chapter 3.3 or the Qiskit Textbook, I'm trying to follow the worked example of the quantum solution for Bernstein-Vazirani. I'm having trouble with what I think should be a trivial bit of ...
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1answer
51 views

On what basis can we write a positive operator as $A=\sum_k\lambda_k|k\rangle\langle k|$?

In Nielsen & Chuang's book equation 2.172 says $$A=\sum_{i}|\widetilde{\psi_i}\rangle \langle \widetilde{\psi_i}| = \sum_j |\widetilde{\phi_j}\rangle \langle \widetilde{\phi_j}|.$$ Then it makes ...
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1answer
46 views

Why is the subscript like this in the equation $\sum_i |\psi_i\rangle \langle\psi_i| = \sum_{ijk} u_{ij} u_{ik}^{*}|\phi_j\rangle \langle\phi_k|$?

In Nielsen's book when proving "Unitary freedom in the ensemble for density matrices"(Theorem 2.6): $$\text{Suppose }|\widetilde{\psi_i}\rangle = \sum\limits_{j}u_{ij} |\widetilde{\phi_j}\...
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1answer
74 views

Why can any density operator be written this way? (quantum tomography)

From page 24 of the thesis "Random Quantum States and Operators", where $(A,B)$ is the Hilbert-Schmidt inner product: \begin{aligned} \rho &=\left(\frac{1}{\sqrt{2}} I, \rho\right) \frac{...
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3answers
103 views

What is the Kraus representation of the quantum channel with Choi $\lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$?

This matrix $$c_{\lambda} = \lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$$ is the Choi–Jamiołkowski matrix of a quantum channel for any $\lambda \in [0,1]$. The ...
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1answer
45 views

Show that $\lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$ is the Choi–Jamiołkowski matrix of a quantum channel

I'm curious how to show how this matrix: $$c = \lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$$ is the Choi–Jamiołkowski matrix of a quantum channel for any $\...
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2answers
60 views

How do you mix two pure states to obtain a mixed state?

If we have the following two states \begin{equation} |\psi\rangle_1 = \frac{1}{\sqrt{2}}|0\rangle_A|0\rangle_B + \frac{1}{\sqrt{2}} |1\rangle_A |1\rangle_B \end{equation} \begin{equation} |\psi\...
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2answers
66 views

How does the sum of two operators act on a two-level system of qubits?

I am confused how the sum of N operators will act on an N-level system of qubits. Here, lets say N=2 so the state is $|00⟩_{CD}$. Then how will this operator $ X_{C} + Z_{D} ⊗ I_{C} + X_{D}$ act on ...
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1answer
151 views

Calculate the von Neumann Entropy of a two-qubit entangled state

After working through an exercise I got a confusion answer/solution that either may be because I've made a mistake or I'm not understanding von Neumann Entropy. I have the two qubit system $$ | \psi \...
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1answer
86 views

How to get a qubit into superposition?

I have a qubit $$\left| \psi \right> = (\alpha_1 + i\alpha_2 ) \left|0\right> + (\beta_1 + i\beta_2 )\left|1\right>$$ so if i give values $\alpha_1 + i\alpha_2 = 1 + 4i$ and $\beta_1 + i\...
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1answer
36 views

Textbook 2.5 (Qiskit) - Unitary and Hermitian matrices

In section 2.5 of the Qiskit textbook, it states that $X$, $Y$, $Z$ and $H$ are examples of unitary Hermitian matrices. As I understand it, this means that the following rule applies: $$UU^\dagger=U^\...
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3answers
40 views

How to prove that the transpose operation maps an arbitrary qubit to its complex conjugate?

How to prove that the transpose operation maps an arbitrary qubit to its complex conjugate, $|\psi^*\rangle \rightarrow |\psi\rangle$
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1answer
78 views

How does a general rotation $R_\hat{n}(\theta)$ related to $U_3$ gate?

From eqn. $(4.8)$ in Nielsen and Chuang, a general rotation by $\theta$ about the $\hat n$ axis is given by $$ R_\hat{n}(\theta)\equiv \exp(-i\theta\hat n\cdot\vec\sigma/2) = \cos(\theta/2)I-i\sin(\...
5
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2answers
93 views

How do I decompose the given $4\times 4$ matrix in terms of Pauli matrices? [duplicate]

I have been working on a question where I have to decompose this matrix in terms of Pauli Matrices: \begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\...
2
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1answer
68 views

Measurement probability of a state from three hilbert spaces

I'm curious how to find the probability measurement of a state when one qubit is measured. For example this state: $|\gamma\rangle = \frac{1}{\sqrt{2}}(| 010 \rangle + | 101 \rangle )$ Assuming these ...
2
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1answer
52 views

Why does $\sum_n \langle n|M_m\rho M_m^\dagger|n\rangle$ simplify to $\langle \psi|M_m^\dagger M_m|\psi\rangle$?

I was trying to derive the formula for $p(m)$ in exercise 8.2 on page 357 in Nielsen & Chuang. But I am wondering what rule I can apply to simplify this $$\mathrm{tr}(\mathcal{E}_m(\rho) )= \...
2
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1answer
83 views

Is the order of the tensor product in $|\phi\rangle\otimes|\chi\rangle=|\chi\rangle\otimes|\phi\rangle$ relevant? [duplicate]

I am reading this book “Quantum Computing Explained” by David McMahon. I found the following statement on page 74 Note that the order of the tensor product is not relevant, meaning $|\phi\rangle \...
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1answer
70 views

Bell's experiment: checking the probabilities

I have some doubts on the calculations behind Bell's experiment, which goes as follows: a pair of entangled photons is shared between two scientists, Alice and Bob, who possess a bunch of polarizers ...
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2answers
48 views

Phase Kickback - factoring Dirac representation

In section 2.3 of the Qiskit textbook (Phase Kickback), there's an example where a controlled-T gate is applied to $|1+\rangle$. You're asked to attempt the same thing with $|0+\rangle$. I've done ...
4
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1answer
28 views

Does $\frac{\Pi_A\otimes I_B}{\text{Tr}((\Pi_A\otimes I_B)\rho_{AB})}\rho_{AB}=\rho_{AB}$ hold for a state $\rho_{AB}$ and projector $\Pi_A$?

For some projector $\Pi_A$ and state $\rho_{AB}$, let $$\sigma_{AB} = \frac{\Pi_A\otimes I_B}{\text{Tr}((\Pi_A\otimes I_B)\rho_{AB})}\rho_{AB}$$ Is it the case that $\sigma_B = \rho_B$? It seems ...
2
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1answer
50 views

Show algebraically that $U|0\rangle\otimes |0\rangle+U|1\rangle\otimes|1\rangle=|0\rangle\otimes U^T|0\rangle+|1\rangle\otimes U^T|1\rangle$

Suppose that Alice applies a unitary operator $U$ with real entries to her qubit in an EPR pair $|\beta\rangle=\frac{1}{\sqrt 2}(|00\rangle+|11\rangle)$. Is this the same as having Bob apply $U^T$ to ...
2
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1answer
39 views

How would I compute a density matrix of a complex qubit mixed state?

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed state, $$ \frac{1}{9}\begin{bmatrix} 5 & 1 & −i \\ 1 & 2 &...
2
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1answer
46 views

How would I compute a density matrix of a 2 qubit mixed state?

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed states, how would I do this? $$ |00> \;with \;probability \; 2/4 \\ ...
2
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1answer
39 views

How do I represent my 3-qubit state in the computational basis?

I have taken the tensor product of $|0\rangle \otimes |-\rangle \otimes |+\rangle$ which resulted in the matrix $$\begin{bmatrix} 1/2\\ 1/2 \\ -1/2 \\ -1/2 \\ 0 \\ 0\\ 0\\ 0\\ \end{bmatrix}.$$ How ...
0
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1answer
44 views

Qiskit textbook - exercise 2.2 - why doesn't the correct answer give the same statevector?

In the Qiskit textbook, I'm stuck on one of the exercises in section 2.2. The question is: Write the state: as two separate qubits. The answer has already been covered in this question here, and is ...
0
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1answer
23 views

How many real numbers are required to describe density matrix for $n$ qubits?

(All of these coming from the topic of simulation of quantum systems) A density matrix $\rho$ Which describe state of $n$ qubits will have $2^{n} \times 2^{n}$ size. We have couple of conditions like ...
1
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1answer
45 views

Partial Cyclic Permuation with only Toffoli and CNOT gates?

I have been trying to solve this puzzle of constructing this transformation from CNOT and Toffoli gates as mentioned in NC page 193 (Ex 4.27) Here is what I have done: First observation is it has 8 ...
0
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1answer
72 views

Derivation of Equation $8.7$ in Nielsen Chuang [duplicate]

Equation \eqref{eq:sp1} represents the reduced state of the system after tracing over environment.(Page number 358) $$\mathcal{E}(\rho) = \mathrm{tr}_{env}(\lbrack U(\rho \otimes \rho_{env} )U^{\...
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0answers
38 views

What does “bipartite system” mean? [duplicate]

I have seen this term appearing multiple times when discussing density matrices. For example here is an excerpt from the lecture notes: We are now ready to introduce the idea of a reduced density ...
1
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1answer
90 views

Is there a different way to represent Pauli gates in X basis?

It's easy to see that in computational basis, Pauli matrices could be represented in the outer product form: $$ X=|0\rangle\langle1|+|1\rangle\langle0|\\ Y=-i|0\rangle\langle1|+i|1\rangle\langle0|\\ Z=...

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