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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What would be the meaning of an $i$ in a qubit state $i\alpha|0\rangle+\beta|1\rangle$?
I do not know if the question is not too easy, but I'll put it here, because I'm interested in it.
So the state of a qubit is often stated in this form:
$$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$...
- 678
4
votes
1
answer
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Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem
In Nielsen and Chuang, in the Fidelity section, (Lemma 9.5, page 410 in the 2002 edition), they prove the following.
$$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$
...
- 1,631
4
votes
2
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In the proof of the joint entropy theorem, why are $p_i\lambda_i^j$ the eigenvalues?
From section 11.3.2 of Nielsen & Chuang:
(4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|...
- 535
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2
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How to show a density matrix is in a pure/mixed state?
Say we have a single qubit with some density matrix, for example lets say we have the density matrix $\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}$. I would like to know what is the ...
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1
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How do I calculate the von Neumann entropy of a pure one-qubit density matrix?
Let's say I have a pure state of the form:
$$\psi = \sqrt{\frac{3}{9}} \lvert 0 \rangle + \sqrt{\frac{6}{9}} \lvert 1 \rangle$$
Then the density matrix representation would be:
$$\rho = \psi \otimes \...
- 1,785
6
votes
2
answers
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How to translate matrix back into Dirac notation?
In Circuit composition and entangled states section of Wikipedia's article on Quantum logic gates the final result of a combined Hadamard gate and identity gate on $|\Phi^{+}\rangle$ state is:
$ M \...
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1
answer
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General parametrisation of an arbitrary $2 \times 2$ unitary matrix
From Nielsen & Chuang's Quantum Computation and Quantum Information (QCQI):
Since $U$ is unitary, the rows and columns of $U$ are orthonormal, form which it follows that there exist real numbers $...
- 603
6
votes
2
answers
872
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How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?
In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation:
$$\mathcal E(\rho) = \mathrm{Tr}_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$...
- 1,631
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2
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How to find the operator sum representation of the depolarizing channel?
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 ...
- 3,085
4
votes
1
answer
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Can vectorization lead to mixed states?
Given an operator $L = \sum_{ij}L_{ij}\vert i\rangle\langle j\vert$, in some basis, the definition of vectorization is $vec(L) = \sum_{ij}L_{ij}\vert i\rangle\vert j\rangle$. The operation is ...
- 3,085
14
votes
3
answers
6k
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Density matrix after measurement on density matrix
Let's say Alice wants to send Bob a $|0\rangle$ with probability .5 and $|1\rangle$ also with probability .5. So after a qubit Alice prepares leaves her lab, the system could be represented by the ...
- 1,785
6
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1
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Confusion on the definition of the phase-damping channel
I am reading about the phase damping channel, and I have seen that some of the different references talking about such channel give different definitions of the Kraus operators that define the action ...
7
votes
2
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882
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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. ...
- 173
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2
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Is the Kraus representation of a quantum channel equivalent to a unitary evolution in an enlarged space?
I understand that there are two ways to think about 'general quantum operators'.
Way 1
We can think of them as trace-preserving completely positive operators. These can be written in the form
$$\...
17
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3
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What does "measurement in a certain basis" mean?
In the Wikipedia article about Bell states it is written:
Independent measurements made on two qubits that are entangled in Bell states positively correlate perfectly, if each qubit is measured in ...
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