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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Nielsen and Chuang: Demonstration of equation 2.12

First, I apologize if something is poorly written but English is not my first language. Reproduced from Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition) in ...
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36 views

Step-by-step passages in calculation

I would like to better understand some passages in a paper (Appendix A): Properties of Tensor Product Bilinearity: $A\otimes(B+ C) = A \otimes B + A \otimes C $ Mixed-product property: $(A\otimes B)(...
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1answer
48 views

Problem about entanglement swapping

I have been learning about the concept of entanglement swapping and found an equation mentioned in the textbook, Mathematics Of Quantum Computing: An Introduction written by Wolfgang Scherer. At ...
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1answer
20 views

Why does the entanglement negativity equal (in magnitude) the sum of the negative eigenvalues?

The entanglement negativity, introduced in (Vidal and Werner 2002), is defined as $$\mathcal N(\rho) \equiv \frac{\|\rho^{T_B}\|_1-1}{2}.$$ It is mentioned there that this equals the sum of the ...
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2answers
114 views

How to find the matrix representation of an operator from its action on a basis?

First, I apologize if something is poorly written but English is not my first language. I know that these exercises have been solved in this question. But I do not agree. Inner product and concrete ...
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1answer
77 views

How to apply the Schmidt Decomposition to a Bell state?

I am trying to understand the Schmidt Decomposition, currently in my QC class. We had a tutorial where we were told if $|\psi\rangle$ is a pure state of a composite system A then there exists $|i_A\...
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1answer
36 views

Why is the function $f_s(x)=\sum_i x_i s_i \pmod 2$ balanced?

A parity function $f_s:\{0,1\}^{n}\rightarrow\{0,1\}$, for some $s\in \{0,1\}^n$, is a function of the form $f_s(x) = x \cdot s$, where the inner product is taken modulo 2. Show that $f_s$ is a ...
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1answer
39 views

Can $U(\rho_1\otimes\rho_2)U^\dagger$ be entangled if either of $\rho_1$ or $\rho_2$ is a maximally mixed state?

Given two qubit states $\rho_1$ and $\rho_2$. By applying some unitary $U$ we get $\rho = U(\rho_1 \otimes \rho_2)U^\dagger$. Can $\rho$ be entangled if either of $\rho_1$ or $\rho_2$ is a maximally ...
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3answers
169 views

How to spot the matrix representation of the quantum NOT operation

Applying the above construction to AND we get the map $(x1,x2,y) \rightarrow (x1,x2,y⊕(x1∧x2))$ for $x1,x2,y \in \{0,1\}$. The unitaryoperator which implements this is then simply the map $|x1〉|x2〉|y&...
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3answers
51 views

Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates

Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates and write down the corresponding circuit. Recall from Quantum Information Theory that $Z=HXH$. As $CNOT$ is a ...
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2answers
26 views

Find the probability of a measurement outcome in terms of the coefficients of the state

Suppose we have a quantum state $|\psi \rangle$ of $n$ qubits, where $|\psi\rangle=\sum_{x∈\{0,1\}^n}\alpha_x |x\rangle$,and we measure the first qubit of $|\psi\rangle$ in the computational basis. ...
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1answer
47 views

Understanding global phase

I have looked at the following: What is the difference between a relative phase and a global phase? In particular, what is a phase? Global and relative phases of kets in QM Global phases and ...
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55 views

Example of Simple phase Change Using 2 Qubits [closed]

I need to manipulate phases in 2 qubits. I will eventually create 34 distinct phase sets to map to an alphabet I built. I see https://github.com/oreilly-qc/oreilly-qc.github.io/blob/master/samples/...
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156 views

Procedures and intuition for designing simple quantum circuits?

I'm working my way through one of the quantum circuits sections in Nielsen and Chuang and I'm struggling to get a feel for the basics of circuit construction. For example, one of the exercises is as ...
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0answers
28 views

How do you decompose an arbitrary quantum state into its corresponding projection subspaces such that their direct sum is the quantum state?

I understand that every Hilbert space $H$ can be decomposed into two mutually orthogonal subspaces $H_1$ and $H_2$ whose direct sum is $H$. Therefore, every vector $v\in H$ can be decomposed into $...
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1answer
50 views

What is the physical meaning of the Hamiltonian $H = \alpha ( |01 \rangle \langle10| + | 10 \rangle \langle 01| )$?

In natural basis $| 0 \rangle = \begin{pmatrix} 1 \\0 \end{pmatrix}$, $| 1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, what physical situation/model does the following Hamiltonian represent: $H = ...
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1answer
29 views

Quantum operation to get rid of small but nonzero eigenvalues

Updated and edited question: Let $N_{\delta}:P(\mathcal{H}_A)\rightarrow P(\mathcal{H}_B)$ be a completely positive trace nonincreasing map from the set of positive semidefinite operators in $\...
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1answer
77 views

Unable to use qiskit_textbook module

When I tried to use the Qiskit textbook module, it threw the following error: Command was: ...
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2answers
70 views

What happens when you send a Bell state through depolarizing channel?

For noise parameter $Q$ and a density matrix $\rho$, we know that the depolarization channel $\mathcal{E}$ would act like: $$ \mathcal{E}(\rho) = (1 - Q)\rho +Q\frac{I}{2}, $$ where $I$ is the ...
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2answers
117 views

How to decompose a unitary single qubit gate by universal quantum gate set?

How to decompose a unitary single qubit gate? I have read some paper or books, which told me a unitary single qubit gate could be decomposed by universal quantum gates set. For example {phase gate, ...
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2answers
58 views

Bipartite states whose coefficients are entries of a unitary matrix

I've been trying to solve this question It seems that in order to show it has unit length, we must show that $$ \frac{1}{d} \sum_{m, n=0}^{d=1} \lvert U_{m, n}\rvert ^2 = 1 $$ I've tried searching ...
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2answers
74 views

What are the possible qubit states?

For a quantum state on the form $$|\psi \rangle = \alpha |0 \rangle + \beta |1 \rangle$$ which possible qubit states can you construct from this? I know that $\alpha$ and $\beta$ must satisfy $$|\...
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2answers
45 views

Find the number of elements in the Schmidt decomposition of a pure state

Consider a pure state $\boldsymbol{\eta} \in \mathcal{H}_{AB}$. There exist orthonormal sets $\{\alpha_1, \alpha_2 \dots \alpha_i\} \subset \mathcal{H}_A$ and $\{\beta_1, \beta_2 \dots \beta_i\} \...
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0answers
39 views

Generalized set of Pauli elements for a basis for the linear transformations on the vector space [duplicate]

I have been doing some practice problems from "Gentle introduction to Quantum Computing". I am a little bit lost with this one: The generalized Pauli group $\mathcal G_n$ is defined by all elements ...
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6answers
139 views

Show that $I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$ for all states $\rho$

I am trying to show that for any qubit state p, the following holds: $$I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$$ I have tried different manipulations,...
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1answer
63 views

How to compute the tensor product of the depolarizing channel with the identity?

Consider two quantum systems A and B, B goes through a depolarizing noise channel, while A is not changed, i.e., they go through the channel $\mathbb{I}_A \otimes \mathcal{E_{\text{depol}}} $. If the ...
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2answers
864 views

Can arbitrary matrices be decomposed using the Pauli basis? [duplicate]

Is it possible to decompose a hermitian and unitrary matrix $A$ into the sum of the Pauli matrix Kronecker products? For example, I have a matrix 16x16 and want it to be decomposed into something ...
3
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1answer
109 views

Estimate of the absolute value of the probability amplitude of |0⟩ in the superposition

You are given a source of an unknown qubit state. You make measurements in the computational basis, that is, your measurement is $\{|0\rangle \langle0|,|1\rangle \langle 1|\}$. You observe that you ...
6
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2answers
210 views

How to translate matrix back into Dirac notation?

In Circuit composition and entangled states section of Wikipedia's article on Quantum logic gates the final result of a combined Hadamard gate and identity gate on $|\Phi^{+}\rangle$ state is: $ M \...
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2answers
1k views

Depolarizing channel operator sum representation

In Nielsen and Chuang (page:379), it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity ...
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2answers
206 views

How do you represent the output of a quantum gate in terms of its basis vectors?

I'm stuck while trying to understand the Hadamard Gate in a more linear algebra understanding. (I understand the algebraic way). This is because I want to program a simulation of a quantum computer. ...