All Questions
Tagged with textbook-and-exercises mathematics
99
questions
0
votes
1
answer
501
views
How to prove that the trace of a density matrix is $1$?
Equation 2 gives the following proof:
$$
\text{Tr}[\rho] = \sum_x \langle x\vert \rho\vert x\rangle = \sum_x \langle x\vert
\sum_i p_i\vert \psi_i\rangle \langle \psi_i\vert\vert x\rangle = \sum_i ...
- 822
1
vote
2
answers
349
views
How to perform a basis change on a 2x2 density operator?
I have an ensemble described by following density operator:
$$
P=3/8 |+\rangle\langle+| + 5/8 |-\rangle\langle-|
$$
I am trying to write this operator in $\{|0\rangle, |1\rangle\}$ basis.
I know that ...
- 47
3
votes
1
answer
194
views
Heisenberg Uncertainty Principle (Nielsen and Chuang Box 2.4)
I'm trying to follow Nielsen and Chuang Book on Quantum Computation and Quantum Information. There is Box 2.4 on the Heisenberg Uncertainty Principle. I got stuck pretty fast. In that box they define:
...
- 177
2
votes
1
answer
211
views
Writing a Density matrix in terms of the magnitude of the Bloch Vector
Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 ...
- 472
1
vote
1
answer
99
views
Do the linear operators $M\otimes I$ and $I\otimes N$ commute?
If not, does that mean that when doing partial measurements on two different shares of an entangled state, the results (expressed as a proability mass function) can depend on the order (i.e who ...
- 159
1
vote
1
answer
215
views
Do unitary matrices acting on entangled states always give a quantum state?
I'm trying to understand what happens when Alice(Bob) apply a unitary to her(his) part of an entangled state. Let us consider the following unitary transformations:
$$U_1 = \frac{1}{\sqrt{2}}
\...
- 11
2
votes
1
answer
1k
views
What are the eigenstates of an operator?
Sorry if this is a silly question, I am new to quantum computing
I was just reading this article that talked about the eigenstates of an operator. And I wonder, how can we find those eigenstates for a ...
- 21
1
vote
1
answer
97
views
Verify that $\langle \sigma^x\rangle^2+\langle\sigma^y\rangle^2+\langle\sigma^z\rangle^2=1$ for $|\psi\rangle=\cos\theta|0\rangle+\sin\theta|1\rangle$
I am trying to solve an exercise, but I can't seem to get it to work.
I get given this rule,
$$\langle \sigma_x \rangle^2 + \langle \sigma_y \rangle^2+\langle \sigma_z \rangle^2 =1 $$
and I am asked ...
- 103
4
votes
3
answers
4k
views
How to represent the Hadamard gate as a rotations on the Bloch sphere?
I am new to Quantum Computing, and I have decided to try and learn the quantum gates. I am trying to understand how to represent some basic gates as rotations on the Bloch Sphere. I was able to ...
- 165
5
votes
1
answer
695
views
Showing that two unitary matrices are equal up to a global phase
Let $U$ and $V$ be two $d × d$ unitary matrices, representing two reversible quantum processes
on a $d$-dimensional quantum system. We say that the two processes “act in the same way”
on the state $|ψ\...
- 373
2
votes
2
answers
180
views
How to prove that the trace of n-qubit matrices satisfies ${\rm Tr}(XY)=2^n\sum_{M\in\{I,X,Y,Z\}^n} x_M y_M$?
It is known that for n-qubit matrices X, Y $\in \mathbb{C}^{2^{n}\times 2^{n}}$ (and Pauli matrices $I, X, Y, Z$) such that
$$
X = \sum_{M \in \{I, X, Y, Z\}^{n}} x_{M}M_{1}\otimes ... \otimes M_{n}
$...
- 163
3
votes
2
answers
143
views
What does $ A - \langle A \rangle $ mean?
I've seen the uncertainty of $A$ written as $$ (\Delta A)^2 = \langle (A - \langle A \rangle)^2 \rangle. $$ But what does this even mean since $ A $ is an operator and $ \langle A \rangle $ is a ...
- 611
1
vote
3
answers
158
views
How to show that the QFT satisfies $\frac1{\sqrt N}\sum_j\prod_le^{2\pi i j_l k/2^l}|j_1...j_n⟩=\bigotimes_l \frac1{\sqrt2}(|0⟩+e^{2\pi i k/2^l}|1⟩)$?
I'm reading Ronald de Wolf's lecture notes, and in chapter 4.5 he writes that
$$
\frac{1}{\sqrt N}\sum\limits_{j=0}^{N-1}\prod\limits_{l=1}^{n}e^{2\pi i j_l k / 2^l}|j_1...j_n\rangle =
\bigotimes\...
- 141
2
votes
1
answer
111
views
how to obtain partial transpose of a Tripartite operator?
i know for a bipartite system with elements
|ij><kl|
elements of its partial transpose are
|kj><il|
now suppose a ...
- 761
1
vote
1
answer
162
views
Two-qubit Bell measurement matrix where the two qubits are not contiguouis
In the answer here, it is explained that where the measurement operates on only a subset of the qubits of the system (for example qubits 2 and 3 out of five), the matrix can be constructed using the ...
- 701