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.
Questions tagged [textbook-and-exercises]
724 questions
20
votes
3 answers
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 the relevant basis.
What does it even mean to…
user72
15
votes
3 answers
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 a set of probabilities corresponding to states…
C-Roux
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14
votes
3 answers
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 following density matrix: $$\rho = .5 |0\rangle…
QuestionEverything
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13
votes
1 answer
General parametrisation of an arbitrary $2 \times 2$ unitary matrix
From Nielsen & Chuang's Quantum Computation and Quantum Information (QCQI):
Since $U$ is unitary, the rows and columns of $U$ are orthonormal, form which it follows that there exist real numbers $\alpha$, $\beta$, $\gamma$ and $\delta$ such that $$…
Tech Solver
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11
votes
2 answers
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 matrix with
$$\frac{\mathbb{I}}{2} = \frac{\rho +…
user1936752
- 3,311
- 1
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- 24
11
votes
3 answers
Is the tensor product of two states commutative?
I'm reading "Quantum Computing Expained" by David McMahon, and encountered a confusing concept.
At the beginning of Chapter 4, the author described the tensor product as below:
To construct a basis for the larger Hilbert space, we simply form…
akawarren
- 111
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- 5
10
votes
2 answers
How does the spectral decomposition of the Choi operator relate to Kraus operators?
In Nielsen and Chuang's QCQI, there is a proof states that
Theorem 8.1: The map $\mathcal{E}$ satisfies axioms A1, A2 and A3 if and only if
$$
\mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger}
$$
for some set of operators…
Sherlock
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10
votes
2 answers
Is the Kraus representation of a quantum channel equivalent to a unitary evolution in an enlarged space?
I understand that there are two ways to think about 'general quantum operators'.
Way 1
We can think of them as trace-preserving completely positive operators. These can be written in the form
$$\rho'=\sum_k A_k \rho A_k^\dagger \tag{1}$$
where…
Quantum spaghettification
- 1,532
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10
votes
2 answers
Procedures and intuition for designing simple quantum circuits?
I'm working my way through one of the quantum circuits sections in Nielsen and Chuang and I'm struggling to get a feel for the basics of circuit construction. For example, one of the exercises is as follows:
This exercise seems really simple on the…
Arthur Allshire
- 101
- 3
9
votes
4 answers
How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?
I am trying to solve this exercise from Qiskit's textbook, problem set 2 "Basic Synthesis of Single-Qubit Gates":
Show that the Hadamard gate can be written in the following two forms
\begin{equation}
H = \frac{X + Z}{\sqrt{2}} \equiv \exp\left(i…
walid
- 335
- 1
- 9
9
votes
1 answer
How are arbitrary $2\times 2$ matrices decomposed in the Pauli basis?
I read in this article (arXiv) Appendix III p.8, that for $A\in \mathcal{M}_2$, since the normalized Pauli matrices $\{I,X,Y,Z\}/\sqrt{2}$ form an orthogonal matrix basis.
$$A=\frac{Tr(AI)I+Tr(AX)X+Tr(AY)Y+Tr(AZ)Z}{2} $$
I don't understand, where…
lufydad
- 491
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9
votes
2 answers
What does it mean to "measure an operator"?
I was reading a book and then I found this statement. I will put the text as well as a screenshot of the text.
The expectation value of an operator is the mean or average value of that operator
with respect to a given quantum state. In other words,…
user27286
- 1,015
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8
votes
2 answers
Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$
I've been trying to figure this out for a while and I'm totally lost. My goal is to show that for two density operators $p$, $q$, that $$\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$$
So far I have constructed a weak inductive hypothesis over n…
8
votes
2 answers
How to show a density matrix is in a pure/mixed state?
Say we have a single qubit with some density matrix, for example lets say we have the density matrix
$$\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}.$$
I would like to know what is the procedure for checking whether this state is pure or…
bhapi
- 919
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- 17
8
votes
1 answer
Is the set of classical-quantum states convex?
I read about the classical-quantum states in the textbook by Mark Wilde and there is an exercise that asks to show the set of classical-quantum states is not a convex set. But I have an argument to show it is a convex set. I wonder whether I made a…
qquery
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