All Questions
16
questions
1
vote
0
answers
104
views
Power rule for derivative operator $D^n$
I was playing with commuting the derivative operator $D$ and the scalar multiplication of $x$ applying to a function $f$. If $[\cdot,\cdot]$ denote de commutator, then:
$$[D,x]f=D(xf)-xDf=f+xDf-xDf=f$$...
- 495
3
votes
1
answer
130
views
Commutator bertween shift operator and "position" operator
I was studying the shift operator in quantum mechanics from wikipedia and in the commutator section explains that for a shift operator $\hat T(x)$ and a position operator $\hat x$, the commutator $\...
- 495
2
votes
1
answer
370
views
If $\hat{A}$ commutes with $\hat{r}^2$, does $\hat{A}$ commute with $\hat{r}$?
Let $\hat{x}, \hat{y}, \hat{z}$ be the position operators in 3D space and $\hat{r}^2=\hat{x}^2+\hat{y}^2+\hat{z}^2$ as usual.
If one operator $\hat{A}$, commutes with $\hat{r}^2$, i.e.
$$\Big[\hat{A}, ...
- 51
1
vote
2
answers
49
views
Algebra involving operator of x and p
For operators x and p in QM with $p= {h\over i}{d \over dx} $, how can I find the combination of operator such as
$$(xp-px)^2 $$ or $$(x+p)(x-p) $$
Can I just expand them by using normal algebra such ...
- 13
0
votes
1
answer
92
views
Is this positive map also completely positive?
Can $T$ defined below be shown to be a completely positive map?
$$T(A)=E[(I-XX')A(I-XX')]$$
$T$ maps between covariance matrices, X is a (column)-vector sampled from some $d$-dimensional distribution (...
- 5,477
3
votes
1
answer
134
views
Mathematical operations order when using an operator
I am not very familiar with operators (as I do not study mathematics) and I have just started a Quantum Mechanics course in a university. However, I am not sure what should be the precise order of ...
- 87
3
votes
2
answers
354
views
Extension of Choi-Jamiolkowski isomorphism to not completely positive maps
In quantum physics, the concept of Channel-State duality is of prime importance in understanding the final state of the system after passing through a channel or performing quantum operations. The ...
- 587
10
votes
1
answer
300
views
Is the failure of $\mathcal{B}(H)\simeq H\otimes H^*$ in infinite dimensions the reason for non-normal states in quantum information?
In the algebraic formulation of quantum physics/information, states $\omega: \mathcal{A}\rightarrow \mathbb{C}$ are defined as linear functionals on a $C^*$-algebra $\mathcal{A}$ (algebra of ...
- 211
3
votes
1
answer
342
views
Trace of a matrix exponential with tensor products, and Von Neumann entropy
$\def\T{\operatorname{Tr}}$
$\def\1{\mathbb{1}}$
Let $H=H_1\otimes H_2\otimes H_3$ be a finite dimensional Hilbert space, and let $\rho_{123}$ be a self-adjoint matrix with $\rho_{123}\geq 0$ (...
- 2,889
2
votes
1
answer
626
views
Integral representation of log of operators
$\def\1{\mathbb{1}}$
Suppose we're in a "good enough" (finite for example) space, and we have positive (semi)-definite operators $P$ and $Q$.
Let $\log{(P)}$ and $\log{(P)}$ be logarithms of $P$ and ...
- 2,889
1
vote
1
answer
31
views
$\left [(A- \langle A \rangle ),\right (B- \langle B \rangle )]=\left [A,B \right ] $
Let A,B two hermitian operator
I don't grasp one of the last step of the proof of Uncertainty principle.
Why:
$$\left [(A- \langle A \rangle ),\right (B- \langle B \rangle )]=\left [A,B \right ] $$
I'...
- 769
2
votes
2
answers
91
views
An equality in linear algebra and operator algebra
Suppose $\rho$ is a self-adjoint postive semidefinite operator in $\mathbb{C}^{n \otimes n}$ (a quantum state). $L$ and $M$ are complex $m \times n$ matrices. How would one prove that if $L^\dagger L \...
- 21
0
votes
0
answers
117
views
Finding the eigenvalues and eignevectors of this operator.
The operator
$$\mathcal{\hat{G}} = (\xi - 1) \sum_{j=1}^N \int dk_j \; k_j \hat{a}^\dagger_j(k_j)\hat{a}_j(k_j),$$
is physically similar to the momentum operator in quantum mechanics. It has the ...
- 403
3
votes
0
answers
220
views
Pureness of Vector States
How does one show that irreducibility is equivalent to a vector state being pure?
In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let $\...
- 802
1
vote
3
answers
142
views
looking for help with a trace/norm inequality
I'm trying to understand a derivation that seems to claim that $\left\vert\text{Tr}\left[\rho U^\dagger\left[U,O\right]\right]\right\vert\leq\|\left[U,O\right]\|$, where $\rho$ is Hermitian and has ...
- 484