Questions tagged [unitary-matrices]

Question 1

I have often read the following statement: Let $G$ be a connected, simple, non-compact Lie Group of dimension $n \geq 2$. Let $ρ: G \to U(H)$ be a unitary representation of $G$ on the Hilbert Space $H$...

Question 2

We were learning quantum computation when we came up with this question that what kind of quantum gates preserve entanglement. We studied the simplest 2-qubit case. It is obvious that quantum gates ...

Question 3

The following question arose while attempting to construct a counterexample to a previous question. A matrix $A$ is said to be unitarily reducible if there exists a unitary matrix $U$ such that $U^tAU$...

Question 4

Consider a general SU(3) Lie algebra element $$ H^{SU(3)} = \left( \begin{array}{ccc} c_3+\frac{c_8}{\sqrt{3}} & c_1-i c_2 & c_4-i c_5 \\ c_1+i c_2 & \frac{c_8}{\sqrt{3}}-c_3 & c_6-i ...

Question 5

I have an integral to solve which arises from solving a quantum eigenvalue problem for a special unitary matrix raised to an $n$-th power. The integral is $$\frac1{\pi}\int_0^{\pi} \cos(2n \alpha) d\...

Question 6

Question: Suppose I have two unitaries: $U_1 = e^{-iA_1} \cdots e^{-iA_N}$, $U_2 = e^{-iA_1}e^{-iB_1} \cdots e^{-iA_N}e^{-iB_N}$, and an ideal unitary $U_{\text{id}} = e^{-i\sum_{k=1}^N A_k}$, where ...

Question 7

Context I am studying Wigner's theorem [1]. I am familiar with unitary operators. I know that for an unitary operator $\widehat{U}$, its inverse equal to its Hermitian adjoint. In other words for a ...

Question 8

$\underline{\text{Problem:}}$ If H is a Hilbert space over $\mathbb{C}$ and $U,V\in \mathscr{B}(H)$ are unitary with $UV=cVU$, where $c\in \mathbb{C}$ and $|c|=1$, then either H is infinite ...

Question 9

I am working with the Lawrence-Krammer representation of $B_n$ and need to find a way to determine if, given any two matrices $A, B$ in the image of the representation, there exists $k\in\mathbb{Z}^+$ ...

Question 10

$\def\lie#1{\mathfrak{#1}}\def\Lie#1{\mathrm{#1}}\DeclareMathOperator{\span}{span}$I'm reading Vandoren and van Nieuwenhuizen's "Lectures on instantons", it is a physicists' review on ...

Question 11

I am interested in computing explicitly representations of the unitary group $U(n)$, which means that given a character in $n$ variables $\chi = \sum_{\lambda \in \mathbb{Y}, l(\lambda) \leq n} a_\...

Question 12

Consider a complex vector $\mathbf{s} \in \mathbb{C}^{N \times 1}$ that is multiplied by a known unitary matrix $\mathbf{G} \in \mathbb{C}^{N \times N}$, yielding $\mathbf{y} = \mathbf{G}\mathbf{s}$. ...

Question 13

Let $A=[a_{ij}],B=[b_{ij}]\in \mathcal{M}_{n}(\mathbb{R})$ be two invertible matrices and additionally, their Hadamard product $A\odot B$ is also invertible. Consider the basic circulant permutation ...

Question 14

For any square complex matrix $A$, there exists a diagonal matrix D and unitary matrix $U,V$ such that $A=UDV$. This question is very general and I almost have no idea about it, cause it didn't give ...

Question 15

The following question has arisen in the course of a physics problem. Suppose I have two Hermitian matrices, denoted by $A$ and $B$. In the problem, I have found two distinct unitary matrices $U_1$ ...

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Questions tagged [unitary-matrices]

Unitary representations of non-compact Lie Groups

What is an equivalent condition for quantum gates to preserve entanglement?

Products of commuting reducible matrices

Eigenvectors of a general SU(3) Lie algebra element

Show that an integral of Chebyshev polynomial yields a Bessel function

Confusions on the error between ideal unitary and product of exponentials

For an antiunitary operator, is inverse equal to its Hermitian adjoint (i.e., $\widehat{U}^{-1} = \widehat{U}^\dagger$)?

Unitary Equivalence of Hilbert space Operators

Algorithm to determine if $A = B^k$ for any $k\geq 0$ if $A, B$ are special unitary matrices

$\mathfrak{su}(2)$ representations inside $\mathfrak{su}(N)$

Is there a standard algorithm to recover a representation of $U(n)$ from its character?

How to find a part of input vector based on a unitary matrix and a part of the input vector?

Can the unitary transformation of one of the matrix preserve the invertibility of Hadamard product? [closed]

For any square complex matrix $A$, there exists a diagonal matrix D and unitary matrix $U,V$ such that $A=UDV$.

Non-uniqueness of the unitary transformation

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