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.
675
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How do I prove the following maps are completely positive?
I am trying to prove that the following superoperators are quantum channels, that is completely positive and trace-perserving linear maps
1 $\Psi[M]=WMW^\dagger$ where $W$ is an isometry
2 $\Psi[M_A]=...
- 137
2
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2
answers
207
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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: $$\text{Tr}(M |\psi\rangle \langle\psi)=\langle\psi| M |\psi\...
- 155
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2
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Performing a projective measurement, is the resulting expectation value $\langle \Psi|M|\Psi\rangle$ bounded between $+1$ and $-1$?
Suppose we have a quantum state $|\Psi\rangle = \alpha|0\rangle + \beta|1\rangle$.According to a measurement operator M, the projective measurement of $|\Psi\rangle$ is given by $\langle\Psi|M|\Psi\...
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1
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help understanding gate to hamiltonian and representation
So I have this question: Given an operator, find some Hamiltonian implementing this operator/gate. I have realized that this is a swap gate and I know the matrix for it. I also know that $U = \text{...
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2
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1
answer
59
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When are two Hermitian operators unitarily similar?
Let $A$ and $B$ $2^n \times 2^n$ Hermitian matrices. What are sufficient and necessary conditions that they are equal up to some unitary, i.e. there exists $U$ such that $A = U B U^\dagger$?
The first ...
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1
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1
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66
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How to take partial trace of a $n - 1$ qubit subsystem from a $n$ qubit system
I would like to calculate the expression
$$
\text{Tr}_2\left\{R^z \sigma\right\}\,,
$$
where
$$
\sigma = \rho \otimes |0\rangle \langle0|^{{\otimes}(n-1)}\,.
$$
Here
$$
R = \sum{\theta_m}G_m\,,$$
...
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1
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1
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84
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Quantum teleportation of unknown qubit when the entangled state is not a Bell state
Assume Bob and Alice have two particles with a prior entanglement: $A$ and $B$. The entangled state $|Ψ⟩$ is maximally entangled, and $$|Ψ⟩ = \frac{1}{\sqrt{2}}(|00⟩ + j|11⟩)\,,$$ where $j$ is a ...
- 21
2
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1
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245
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Trace Distance in Bloch sphere, what is the vector of Pauli matrices?
While reading Chapter 9.2.1 Trace distance in "Quantum Computation and Quantum Information," I encountered a question. What is the vector of Pauli matrices referring to?
$$
\vec{\sigma} = (\...
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2
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102
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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 the state
$|\psi\rangle:=A|00\rangle+B|11\rangle$.
If I have another qubit
in the state $|\phi\rangle:=a|0\rangle+b|1\...
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1
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41
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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 $(...
- 631
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140
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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 ...
- 21
2
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1
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237
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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 $\sigma_{n}$ ...
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2
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1
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84
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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 ...
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1
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63
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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?
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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 ...
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