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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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 ...
- 898
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2
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Find the $\theta$ and $\phi$ values on the Bloch sphere corresponding to the state $\frac{1+i}{2}|0\rangle+\frac1{\sqrt2}|1\rangle$
If I have the following state:
$$
\left| \varphi \right>=\frac{1}{\sqrt{2}}\left(\left(\frac{1+i}{\sqrt{2}} \right)\left| 0 \right> + \left| 1\right>\right)
$$
How can I find the $\theta$ ...
- 408
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2
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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
6
votes
1
answer
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What are examples of the correspondence between channels and their Stinespring dilations?
In this post I read that
"quantum measurements are special cases of quantum channels (CPTP maps). Stinespring's dilation states that any quantum channel is realized by partial tracing a unitary ...
- 175
4
votes
4
answers
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Can a Kraus representation act as the identity on any operator?
In the textbook “Quantum Computation and Quantum Information” by Nielsen and Chuang, it is stated that there exists a set of unitaries $U_i$ and a probability distribution $p_i$ for any matrix A,
$$\...
2
votes
2
answers
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What is the result of measuring $\sigma_x$ on the state $|01\rangle+|10\rangle$?
I confused about how to calculate the probabilities and getting a certain result of measuring Bell's states with Pauli matrices as the operator. When you measure something, the state involved would be ...
17
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3
answers
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What does "measurement in a certain basis" mean?
In the Wikipedia article about Bell states it is written:
Independent measurements made on two qubits that are entangled in Bell states positively correlate perfectly, if each qubit is measured in ...
user72
13
votes
1
answer
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General parametrisation of an arbitrary $2 \times 2$ unitary matrix
From Nielsen & Chuang's Quantum Computation and Quantum Information (QCQI):
Since $U$ is unitary, the rows and columns of $U$ are orthonormal, form which it follows that there exist real numbers $...
- 603
9
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Is the Kraus representation of a quantum channel equivalent to a unitary evolution in an enlarged space?
I understand that there are two ways to think about 'general quantum operators'.
Way 1
We can think of them as trace-preserving completely positive operators. These can be written in the form
$$\...
7
votes
2
answers
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Are three POVM measurements on a single qubit physically realizable?
In Nielsen and Chuang Quantum Computation and Quantum Information book section 2.2.6, a POVM of three elements are used to measure a single qubit in order to know for sure whether the state is $|0\...
- 879
6
votes
1
answer
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Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?
I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following:
$$
F(\rho, \sigma) := \...
- 1,785
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3
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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 ...
- 305
3
votes
3
answers
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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 ...
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What to do when the amount of solutions is not known before applying Grovers Algorithm?
When running Grovers Algorithm one has to know how many solutions there are right? When the number of solutions are not known is then what do you do then?
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