All Questions
26
questions
6
votes
1
answer
177
views
Convergence from spectrum of the Choi matrix?
Suppose I have a linear operation of the form $T(C)=\frac{1}{m}\sum_i^m A_i C A_i$ for a set of $m$ symmetric $d \times d$ matrices $A_i$.
I construct Choi matrix below with $e_i$ referring to ...
- 5,477
1
vote
0
answers
43
views
If $A\geq 0$ and $0\leq B\leq C$, then $0\leq \sqrt{B}A\sqrt{B}\leq \sqrt{C}A\sqrt{C}$ [duplicate]
Let $H$ be a complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. We say $A\geq 0$ if $\langle Ax,x\rangle\geq0$ for all $x\in H$. If $A\geq 0$ and $0\leq B\leq C$, then does $0\leq \...
- 461
0
votes
1
answer
70
views
If $0\leq A\leq B$ and $C\geq0$ then does $0\leq AC\leq BC$?
Let $H$ be a complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. We say $A\geq 0$ if $\langle Ax,x\rangle\geq0$ for all $x\in H$. If $0\leq A\leq B$ and $C\geq0$ then does $0\leq AC\...
- 461
1
vote
1
answer
42
views
What can I infer from this scalar product identity?
Suppose I have shown for a single vector $\lvert 0 \rangle$ that
$$\langle 0 \rvert U^\dagger \phi U \lvert 0 \rangle=\langle 0 \rvert \phi \lvert 0 \rangle$$
where $\phi$ is a certain operator and $U$...
- 121
0
votes
2
answers
58
views
Operator norm of X
I've been trying to prove that the operator norm of $\hat{X}\phi(x) = x\phi(x)$ with $\phi(x) \in L^2[0,1]$ is given by:
$$
||\hat{X}|| = \sup_{||\phi(x)||=1} ||\hat{X}\phi(x)||=1
$$
However i havent ...
1
vote
1
answer
204
views
Hermiticity of operator on square integrable functions
I'm trying to solve this problem for my Quantum Mechanics class but its my first time dealing with infinite dimensional spaces so I'm a little lost here. I need to show that the operator defined by
$$
...
0
votes
1
answer
47
views
If $\sum_{x=1}^{|A|} |\psi_x\rangle\langle \psi_x|=I$, then $|\psi_x\rangle\langle \psi_x|$ is a basis projectoer.
Let $\left\{E_{x}\right\}_{x=1}^{|A|}$ is a rank 1 matrix in $\operatorname{Herm}(A)$ which $\sum_{x=1}^{|A|} E_{x}=I^A$. $A$ is our Hilbert space and $I^A$ is the identity matrix.Then I want to show ...
0
votes
1
answer
142
views
Prove convergence of series under trace-norm topology
Let $H$ be a separable Hilbert space, $T(H)$ the set of trace-class operators, and $D(H)\subset T(H)$ the set of density operators (i.e., positive and having trace 1).
For any unit vector $\vert \...
- 1,254
0
votes
1
answer
92
views
Is this positive map also completely positive?
Can $T$ defined below be shown to be a completely positive map?
$$T(A)=E[(I-XX')A(I-XX')]$$
$T$ maps between covariance matrices, X is a (column)-vector sampled from some $d$-dimensional distribution (...
- 5,477
0
votes
0
answers
220
views
Commutator of Linear Operators Identities in Linear Algebra
I would like an explanation about how commutator identities work.
I am doing Shankar's Principle of Quantum Mechanics book, and the first chapter is all about Linear Algebra. He gives to my hands the ...
- 157
2
votes
1
answer
97
views
$\langle u| \langle v| A |u\rangle |v\rangle \geq 0$ for any $|u\rangle \in \mathcal{H}_1$ and $|v\rangle \in \mathcal{H}_2$. Then $A$ is Hermitian?
$\mathcal{H}_1$ and $\mathcal{H}_2$ are complex Hilbert spaces. $A$ is an operator from $\mathcal{H}_1\otimes \mathcal{H}_2$ to $\mathcal{H}_1\otimes \mathcal{H}_2$.
Suppose that $\langle u| \langle ...
- 861
2
votes
0
answers
68
views
Why are operators often written in 3s?
In a lot of my quantum mechanics and linear algebra books, operators are often defined as
$M=IMI$
and many operations on operators are often done in similar fashion, for eg. operator $M$ might be ...
- 151
8
votes
3
answers
523
views
Computing the product $(\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n$
I want to compute the product
$$
(\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n,
$$
for a natural number $n$. For $n$ equal to 0 or 1, the computation is very simple but for such a low number as 2 the brute ...
- 1,589
1
vote
3
answers
3k
views
Proving that every diagonalizable operator is normal
This question is insipired by the proof of the Spectral Theorem in Nielsen and Chuang's book (page 72). It says:
Theorem: Any normal operator $M$ on a vector space $V$ is diagonal with respect to ...
- 1,862
0
votes
1
answer
287
views
How to express a Hermitian operator with a Unitary operator and a diagonal matrix?
I'm stuck in this problem, and is the last one!.
I have a hermitian operator A with its eigen-everything and I have to prove that it can be writen as $UDU^+$ where U is a unitary transformation, $U^+$...
