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

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
57 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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0answers
18 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 ...
0
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2answers
40 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
39 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
32 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
36 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
31 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
32 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
18 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
32 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
53 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
36 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
82 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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2answers
43 views

Is there a simplified formula for the adjoint of the outer product of ket and bra?

I was reading about measurements and got to some operator like this: $$\left| 0\rangle \langle 0\right| $$ Is there any form I can apply when I have to calculate $$ \left( \left| 0\rangle \langle 0\...
1
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1answer
37 views

Is there an easy way to calculate the eigenvalues of the partial transpose of a given matrix? [duplicate]

Consider the state $$|\psi\rangle=(\cos\theta_A|0\rangle+\sin\theta_A|1\rangle)\otimes(\cos\theta_B|0\rangle+e^{i\phi_B}\sin\theta_B|1\rangle).$$ To calculate the $\rho^{T_B}$ I first calculate the $\...
0
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1answer
33 views

5 qubit codewords definition in terms of operators: Mermin

Book: Quantum computer science by David Mermin Chapter:5 Page-118 The 5-Qbit codewords are most clearly and usefully defined in terms of the $M_{i}$ (rather than writing out their lengthy explicit ...
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1answer
32 views

Upper and lower bound of CHSH operator [closed]

How to find the upper and lower bounds of $a_1^k(b_1^k+b_2^k)+a_2^k(b_1^k-b_2^k)$ with $-1 \leq a^k_1,a^k_2,b^k_1,b^k_2 \geq 1$ for the situation $b_1^k+b_2^k>0$ and $b_1^k-b_2^k>0$; $b_1^k+...
0
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2answers
58 views

Find the unitary implementing the transformation $|0\rangle\to\frac1{\sqrt2}(|0\rangle+|1\rangle),|1\rangle\to\frac1{\sqrt2}(|0\rangle-|1\rangle)$ [closed]

I have found a question for finding the Unitary operator for the following transformation: I found the solution as well. But I didn't understand how they got the solution!
0
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1answer
34 views

In classical state discrimination, why does the trace distance quantify the probability of success?

Consider the following task: we are given a probability distribution $p_y:x\mapsto p_y(x)$ with $y\in\{0,1\}$ (e.g. we are given some black box that we can use to draw samples from either $p_0$ or $...
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1answer
49 views

How do I calculate the eigenvalues of the positive partial transpose of this two-qubit state?

How can I calculate the eigenvalues of $\rho^{T_{B}}$ (PPT) of the following state $$ \rho =\frac{1}{2}|0\rangle\langle0|\otimes|+\rangle\langle+|+\frac{1}{2}|+\rangle\langle+|\otimes|1\rangle\langle1|...
5
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3answers
336 views

Why is the transpose of a density matrix positive and trace preserving?

With density matrix $\rho=\sum_{a,b=0}^1\rho_{a,b}|a\rangle\langle b|$ and it's transpose $\rho^T=\sum_{a,b=0}^1\rho_{a,b}|b\rangle\langle a|$. How to confirm that $\rho^T$ is positive and trace ...
4
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1answer
133 views

What is the difference between $|0\rangle+|1\rangle$ and a balanced mixture of $|0\rangle$ and $|1\rangle$? [duplicate]

Suppose I have a quantum state $\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle$. Also I have a mixture of two quantum states $S_{1} = |0\rangle$ and $S_{2} = |1\rangle$. In this mixture $50\%$...
3
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1answer
43 views

Probability of measuring one qubit from the state of two qubits

I am new to quantum information and I am trying to work on some problems but I have confused myself over a qubit problem. I have the state of two qubits $|\psi\rangle_{AB}=a_{00}|00\rangle+a_{01}|01\...
1
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1answer
69 views

Why isn't $\{1,2,3\}$ well ordered? [closed]

I was reading the book "Quantum Computing Since Democritus". "The set of ordinal numbers has the important property of being well ordered,which means that every subset has a minimum ...
3
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1answer
73 views

How do I determine if a given pure two-qubit state is separable?

I'm trying to self-study some topics about quantum computing and I came across a topic of state separability. Talking about that, I wanted to determine separability on the following state (from Qiskit ...
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1answer
119 views

How do I arrange these two-qubit states based on their entanglement?

How can I arrange the following states in decreasing order based on their entanglement? i) $\rho_1 = \frac{1}{2}|0\rangle \langle0|\otimes |+\rangle \langle+| +\frac{1}{2}|+\rangle \langle+|\otimes |...
3
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1answer
50 views

Shouldn't the input state of Deutsh-Jozsa's algorithm look like $|0\rangle^{\otimes n}\otimes |1\rangle$ rather than $|0\rangle^{\otimes n}|1\rangle$?

According to this wikipedia page the initial state in Deutsch–Jozsa algorithm is written as follows: $$|0\rangle^{\otimes n} |1\rangle$$ shouldn't it look like this?: $$|0\rangle^{\otimes n} \otimes |...
3
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1answer
41 views

How do we find the stabilizer generators for the three-qubit bit-flip code spanned by $|000\rangle$ and $|111\rangle$?

In Nielsen & Chuang's book "Quantum Computation and Quantum Information" section 10.5.6, page 467 there is the following statement Consider the familiar three-qubit bit-flip code ...
3
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3answers
66 views

How does one create the unitary sending $|0\rangle$ into a target quantum state?

The Hadamard gate allows us to construct an equal superposition of states. If one wants to construct an arbitrary superposition e.g. $\alpha\vert 0\rangle + \beta\vert 1\rangle + ..$, how does one ...
2
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1answer
52 views

How is $\sum_i\langle i|M|i\rangle$ correlated to $\mathrm{tr}(M)$?

In the book Quantum computation and quantum information, it says to evaluate $tr(A|\psi\rangle\langle\psi|)$ using Gram-Schmidt procedure to extend $|\psi\rangle$ to an orthonormal basis $|i\rangle$ ...
3
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2answers
65 views

Is a projective measurements over a superposition of eigenstates possible?

All observables admit a spectral decomposition in terms of projectors $P_m$ into the eigenspace corresponding to the eigenvalue $m$. So given for example a collection of kets $|0\rangle, |1\rangle,...,...
4
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2answers
210 views

If the eigenvalues of $Z$ are $\pm1$, why are the computational basis states labeled with “$0$” and “$1$”?

The computational basis is also known as the $Z$-basis as the kets $|0\rangle,|1\rangle$ are chosen as the eigenstates of the Pauli gate \begin{equation} Z=\begin{pmatrix}1 & 0 \\ 0 & -1\end{...
2
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2answers
64 views

Why does Grover's algorithm rotate around $|000…0\rangle$?

In general I understand Grover's algorithm, how we can think of two separated state spaces, a space $|\alpha\rangle$ with no solutions and a space $|\beta\rangle$ with only solutions, and how the ...
3
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1answer
102 views

How to construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift?

How can I construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift? I am stuck in this and any help would be appreciated.
5
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4answers
107 views

Derivation of the identity $\sum_j p_j \langle \psi_j|M|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|M\right)$

For measurement, we know $$\langle M \rangle = \sum_j p_j \langle \psi_j|M|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|M\right).$$ My question is, how can we go ...
7
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2answers
364 views

What's the 'physical consistency' in the partial trace scenario?

I'm reading 'Why the partial trace' section on page 107 in Nielsen and Chuang textbook. Here's part of their explanations that I don't quite understand: Physical consistency requires that any ...
1
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2answers
57 views

How do I trace out the second qubit to find the reduced density operator? [duplicate]

I'm doing an exercise to trace out the second qubit to find the reduced density operator for the first qubit: $tr_2|11\rangle\langle00| = |1\rangle\langle0|\langle0|1\rangle$ I'm just wondering if I ...
3
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1answer
75 views

What's the difference between $p(i|m)$ and $p(m|i)$ in measurement?

Suppose we perform a measurement described by measurement operators $M_m$. If the initial state is $|{\psi_i}\rangle$, then the probability of getting result $m$ is $$ \begin{align} p(m|i)=\| M_m|\...
5
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1answer
175 views

What does $M_m |\psi_i\rangle$ mean in the equation $p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$?

I have trouble understanding two equations in the Nielsen & Chuang textbook. Suppose we perform a measurement described by the operator $M_m$. If the initial state is $|\psi_i\rangle$, then the ...
1
vote
2answers
139 views

How to describe the state of a qubit passing through two Hadamard gates? [duplicate]

Describe the state of the qubit at points A, B and C. How does this demonstrate that we need the “ket” (or the vector) representation of qubits, rather than just describing them in terms of ...
1
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1answer
53 views

Probability on measuring bell state in x-basis with pauli operator sigma x [duplicate]

I am confused about how to calculate the probabilities of getting a certain result when measuring a Pauli observable on a Bell state. When you measure an observable the state is projected onto an ...
-1
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1answer
50 views

Uniqueness of Density Operator

I have been reading "Introduction to Quantum Information Science" by Masahito Hayashi, Satoshi Ishizaka,Akinori Kawachi, Gen Kimura and Tomohiro Ogawa; Springer Publication. I'm currently in ...
2
votes
1answer
41 views

Is there an error on Qiskit.org textbook with the superdense coding section?

The textbook on Qiskit.org has When the H-gate is applied to first qubit, it creates superposition and we get the state $|0+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |01\rangle)$ Shouldn't it be: $$|...
3
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1answer
142 views

Application of QFT to Order-finding

In the Nielsen & Chuang book, section 5.3.1 page 226, there is a statement which goes like this:- (statement-1) The quantum algorithm for order-finding is just the phase estimation algorithm ...
4
votes
3answers
103 views

Writing state $ |\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as separate qubits (qiskit textbook)

While going through the IBM qiskit textbook online, I came across the following question in section 2.2: Write the state: $ |\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as two separate ...
2
votes
2answers
39 views

How to find the normalization factor of the eigenvectors of the $\sigma_x$ Pauli gate?

I'm trying to calcaute the eigenstates for the $\sigma_x$ gate, and I can follow the process up to finding eigenvalues $\pm 1$, but I don't understand where the $\frac{1}{\sqrt{2}}$ coefficient comes ...
3
votes
1answer
85 views

Expectation value of operator - python

I am suposed to solve following problem: Calculate the probability $P_n(x > X)$ that a particle in then n-th eigenstate is found at a position with an x-value larger than X. Here it is convenient ...
4
votes
1answer
108 views

How to show that a given mixed two-qubit state is separable? [closed]

A pure state is separable (unentangled) if it can be written as a tensor product of states of each qubit. A mixed state is separable, if it can be written as a probability distribution over separable ...

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