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
questions
18
votes
2
answers
10k
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 ...
- 898
17
votes
3
answers
8k
views
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
14
votes
3
answers
6k
views
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
14
votes
3
answers
6k
views
How to calculate an Expected Value of some operator acting on qubits?
I'm trying to implement the Variational Quantum Eigensolver in Qiskit.
Suppose, I have an operator $A = \sigma_1^z\sigma_2^z$ acting on some two-qubit state $|\psi\rangle$. After a measurement I get ...
- 898
13
votes
1
answer
6k
views
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
10
votes
2
answers
825
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 ...
- 101
9
votes
4
answers
1k
views
How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?
How can I demonstrate on the exponential part equality of the Hadamard matrix:
$$H=\frac{X+Z}{\sqrt2}\equiv\exp\left(i\frac{\pi}{2}\frac{X+Z}{\sqrt2}\right).$$
In general, how can I demonstrate on:
$\...
- 335
9
votes
2
answers
1k
views
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
$$\...
9
votes
2
answers
3k
views
What does it mean to "measure an operator"?
I was reading a book and then I found this statement. I will put the text as well as a screenshot of the text.
The expectation value of an operator is the mean or average value of that operator
with ...
- 975
8
votes
3
answers
2k
views
Is the tensor product of two states commutative?
I'm reading "Quantum Computing Expained" of David McMahon, and encountered a confusing concept.
In the beginning of Chapter 4, author described the tensor product as below:
To construct a ...
- 81
8
votes
2
answers
4k
views
How to find the operator sum representation of the depolarizing channel?
In Nielsen and Chuang (page:379), it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity ...
- 3,085
8
votes
2
answers
679
views
What's the 'physical consistency' in the partial trace scenario?
I'm reading 'Why the partial trace' section on page 107 in Nielsen and Chuang textbook. Here's part of their explanations that I don't quite understand:
Physical consistency requires that any ...
- 2,398
8
votes
1
answer
2k
views
How are arbitrary $2\times 2$ matrices decomposed in the Pauli basis?
I read in this article (Apendix III p.8) that for $A\in \mathcal{M}_2$, since the normalized Pauli matrices $\{I,X,Y,Z\}/\sqrt{2}$ form an orthogonal matrix basis.
$$A=\frac{Tr(AI)I+Tr(AX)X+Tr(AY)Y+...
- 481
8
votes
3
answers
1k
views
How does the spectral decomposition of the Choi operator relate to Kraus operators?
In Nielsen and Chuang's QCQI, there is a proof states that
Theorem 8.1: The map $\mathcal{E}$ satisfies axioms A1, A2 and A3 if and only if
$$
\mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger}
$$...
- 695
8
votes
2
answers
900
views
How to compute the measurement probability in swap test?
The figure of a circuit and the state are as follows.
The final state before the measurement is $|O_{out}\rangle=\frac{1}{2}|0\rangle(|\phi\rangle|\psi\rangle+|\psi\rangle|\phi\rangle)+\frac{1}{2}|1\...
- 639