All Questions
Tagged with textbook-and-exercises unitarity
25
questions
2
votes
2
answers
84
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Show that transformation $U_f: \left| x, y \right\rangle \to \left| x, y \oplus f(x) \right\rangle$ is unitary [duplicate]
I am reading Quantum Computation and Quantum Information by Chuang and Nielsen and they claim that it is easy to show that transformation $U_f: \left| x, y \right\rangle \to \left| x, y \oplus f(x) \...
- 121
2
votes
1
answer
59
views
When are two Hermitian operators unitarily similar?
Let $A$ and $B$ $2^n \times 2^n$ Hermitian matrices. What are sufficient and necessary conditions that they are equal up to some unitary, i.e. there exists $U$ such that $A = U B U^\dagger$?
The first ...
- 525
1
vote
1
answer
154
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Is a linear combination of unitaries unitary?
Suppose you have a pure state $\vert\psi\rangle$. Consider the following operation.
For unitaries $U_1$ and $U_2$, one can take complex numbers $\alpha, \beta$ where $|\alpha|^2 + |\beta|^2 = 1$ and ...
- 3,085
3
votes
1
answer
107
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Why is the operator $M_a |x\rangle= |a \cdot x \pmod{N} \rangle $ unitary?
If $N\geq 2$, $a\in \mathbb{Z}_N$, and $a^r= 1$ for some $r$. Consider the operator $M_a$, which is related to order finding :
$M_a |x\rangle= |a \cdot x \pmod{N} \rangle $ if $x\in \mathbb{Z}_N$
What ...
- 51
1
vote
0
answers
36
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Why does unitary matrix acts only on input qubit state of a vector that is a result of add modulo 2?
Let $|\psi\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k \oplus b\rangle $ so that $|\psi'\rangle = \frac{1}{\sqrt{2}}[\sum_{k=2}^{1}(-1)^{ka}(U_{A}|k\rangle \otimes U_{B}|k \oplus b\rangle)]...
- 518
2
votes
2
answers
237
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Understanding different forms of an arbitrary Unitary transformation in $\mathcal{H}_2$
I'm working to have a greater understanding of the arbitrary unitary transformation matrix when working in the context of the Bloch sphere. At this time I have found several equivalent ...
- 472
1
vote
1
answer
215
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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:
$$U_1 = \frac{1}{\sqrt{2}}
\...
- 11
5
votes
1
answer
695
views
Showing that two unitary matrices are equal up to a global phase
Let $U$ and $V$ be two $d × d$ unitary matrices, representing two reversible quantum processes
on a $d$-dimensional quantum system. We say that the two processes “act in the same way”
on the state $|ψ\...
- 373
-4
votes
1
answer
64
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Quantum Linear Algebra [closed]
Find a 4 x 4 unitary matrix U such that U = eiA. (Possibly up to multiplying by a unit scalar, U is a matrix seen in the course.) Verify your calculation by showing how if U were given, one can obtain ...
- 1
0
votes
0
answers
60
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Unitary Transformations for States with Same Entanglement [duplicate]
$\newcommand{\Ket}[1]{\left|#1\right>}$
I know this has been asked before in another context (How to construct local unitary transformations mapping a pure state to another with the same ...
3
votes
2
answers
430
views
How to construct local unitary transformations mapping a pure state to another with the same entanglement?
$\newcommand{\Ket}[1]{\left|#1\right>}$In Nielsen's seminal paper on entanglement transformations (https://arxiv.org/abs/quant-ph/9811053), he gives a converse proof for the entanglement ...
0
votes
1
answer
415
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Unitary Transformations for Schmidt Decomposition
$\newcommand{\ket}[1]{|#1\rangle}$
Suppose a pure state $\ket{\psi}$ has a Schmidt decomposition given by $\ket{\psi^{SD}}$, which can be obtained via the diagonalization of the reduced density matrix ...
1
vote
1
answer
67
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Why is it not easy to distinguish $U|\psi\rangle$ and $U'|\psi\rangle$ if $\|U-U'\|<\epsilon$?
So I am currently working on an assignment, which is about the induced Euclidian norm
$$
||A||:= \max_{v\in\mathbb{C}^d\text{ s.t. }||v||_2=1} ||Av||_2
$$
for some $A\in\mathbb{C}^{d\times d}$.
For ...
- 13
0
votes
1
answer
148
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Textbook 2.5 (Qiskit) - Unitary and Hermitian matrices
In section 2.5 of the Qiskit textbook, it states that $X$, $Y$, $Z$ and $H$ are examples of unitary Hermitian matrices. As I understand it, this means that the following rule applies: $$UU^\dagger=U^\...
- 167
1
vote
2
answers
239
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Find the unitary implementing the transformation $|0\rangle\to\frac1{\sqrt2}(|0\rangle+|1\rangle),|1\rangle\to\frac1{\sqrt2}(|0\rangle-|1\rangle)$ [closed]
I have found a question for finding the Unitary operator for the following transformation:
I found the solution as well. But I didn't understand how they got the solution!
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