8
votes
Accepted
How are two photons entangled in the Xanadu computer?
QML researcher at Xanadu here.
Our X-series chip produce entangled states by squeezing light and then combining it at beam splitters: those 'cables' are waveguides in a chip, which when they are close ...
- 196
6
votes
Accepted
What are spin-coherent states?
These are also known as SU(2)-coherent states; one original reference is https://doi.org/10.1103/PhysRevA.6.2211.
In a spin system, with states labeled by the eigenvalue of the total angular momentum ...
- 4,569
5
votes
Accepted
Why does joint ground state not change under action of beam splitting unitary operator?
Calculate
$$
\begin{align}
\hat{U}|00\rangle &= \exp\left(-igt(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\hat{a}_2)\right)|00\rangle \\
&= \sum_{k=0}^\infty \frac{(-igt)^k}{k!}(\hat{a}^\...
- 24.2k
4
votes
How is this expression for a GHZ state obtained in the nature paper by Pan et al. (2000)?
First, in order to express $|H'\rangle$, $|V'\rangle$, $|R\rangle$ and $|L\rangle$ in terms of $|H\rangle$ and $|V\rangle$, add and subtract the pair of equations $(2)$ and add and subtract the pair $(...
- 24.2k
4
votes
Accepted
Why is a state Gaussian if and only if its covariance matrix satisfies $\boldsymbol\sigma+i\boldsymbol\Omega\ge0$?
From the definition of the covariance matrix,
$${\sigma}_{ij}=\left\langle \frac{x_i x_j+ x_j x_i}{2}\right\rangle -\langle x_i\rangle\langle x_j\rangle,$$ where we define
$$\boldsymbol{x}=(q_1,p_1,\...
- 4,569
4
votes
Why does joint ground state not change under action of beam splitting unitary operator?
There's more than one way, and I'll suggest two of them here:
Expand $\hat{U}$ using the formula for the Taylor series of an exponential ($e^\hat{A}$) centered around $\hat{A}=\hat{0}$, and then you ...
3
votes
Accepted
Trace distance between thermal states
Since the multimode Fock states $|\mathbf{n}\rangle=|n_1\rangle\otimes |n_2\rangle\otimes\cdots\otimes |n_M\rangle$ form an orthonormal basis, one can treat each state by their probability ...
- 4,569
3
votes
Comparing technical limitations of major quantum computing approaches
For a general overview about hardware and the difficulties they present, I recommend section three of Quantum Computing: An Overview Across the System Stack. Another good introduction would be Quantum ...
- 2,925
3
votes
Accepted
How is measurement performed on a stream of polarized photons?
One idea is to do polarimetry. By using a polarizing beam splitter, the polarization qubit can have each of its polarization components directed to a different detector for photon counting (ideally a ...
- 4,569
3
votes
Accepted
Relation between Wigner quasi-probability distribution and statistical second-moment
You mean something like
$$W_{G}(\mathbf{r}) =\frac{2^{n}}{\pi^{n} \sqrt{\operatorname{Det} \sigma}} \mathrm{e}^{-(\mathbf{r}-\overline{\mathbf{r}})^{\top} \boldsymbol{\sigma}^{-1}(\mathbf{r}-\overline{...
- 1,853
2
votes
Accepted
How do we restrict to a limited number of dimensions, say 3 for qutrits, while using OAM states of light?
As for any platform, one has to choose a suitable $d$-dimensional "computational" subspace. Suitability depends on your application, but generally it means that one should be able to perform ...
- 5,317
2
votes
Shor's Code: Understanding how it satisfies Knill Laflamme Theorem
Strictly, what you have to calculate is that for all $i$ and $j$
$$
\langle 0_L|U_iU_j|1_L\rangle=0
$$
and
$$
\langle 0_L|U_iU_j|0_L\rangle=\langle 1_L|U_iU_j|1_L\rangle.
$$
(I've ignored the ...
- 60.8k
2
votes
Why does joint ground state not change under action of beam splitting unitary operator?
Let $|\psi\rangle$ be an eigenstate of an operator $A$, $A|\psi\rangle=\lambda|\psi\rangle$.
Then
$$e^A |\psi\rangle = \sum_{k=0}^\infty \frac{A^k}{k!}|\psi\rangle = \sum_{k=0}^\infty \frac{\lambda^k}{...
glS♦
- 26.4k
2
votes
Accepted
How to write the covariance matrix of a quantum gaussian state as a function of photon numbers?
Recall that $a=(q+ip)/\sqrt{2}$ in some dimensionless units (Weedbrook might change the units because I think they like $\hbar=2$; I'm using $[q,p]=i$ and $[a,a^\dagger]=1$). We can thus find the ...
- 4,569
2
votes
Accepted
What are the antisymmetric terms in $\sigma_{mn}$ in the expression for the Fisher information?
Antisymmetric means that $\sigma_{nm}=-\sigma_{mn}$. Since the sum ranges over all values of $m$ and $n$, adding an antisymmetric term adds something proportional to
$$|\langle \psi_m^\lambda|\...
- 4,569
2
votes
Accepted
Does the covariance matrix of $\rho_1-\rho_2$ have a simple expression in terms of the individual covariance matrices?
There are many things being overlapped here that likely have no business together. A Gaussian state is uniquely determined by its covariance matrix and its displacement vector, any linear combination ...
- 4,569
2
votes
Accepted
How do the Type I and Type II fusions work?
NOTE:
I've seen this line of reasoning and those calculations somewhere, but I don't remember where. Here is my reconstruction of them.
Type I:
Let's have in mind the sketch of fusion type I from the ...
- 268
2
votes
Accepted
Are there resources to learn about quantum optics that focus on math and/or are addressed to those familiar with the qubit formalism?
There are a number of rather well regarded review papers, such as:
https://doi.org/10.48550/arXiv.quant-ph/0503237,
https://doi.org/10.1016/j.physrep.2007.04.005,
https://doi.org/10.1103/RevModPhys.84....
- 4,569
1
vote
Hong Mandel Effect - question on math of indistinguishability of photons
Your state is perfectly normalized if you use that $(a^\dagger)^2|0\rangle = \sqrt{2}|2\rangle$ (where $|0\rangle$ and $|2\rangle$ are normalized states): Its norm is
$$
(t^2-r^2)^2 + (\sqrt{2}tr)^2 + ...
- 7,683
1
vote
How to compute a partial trace of the form ${\rm Tr}_S [s^\dagger b- s b^\dagger, \vert \psi \rangle \langle \psi \vert \otimes \rho_B]$?
This can be computed directly by using the partial trace $Tr_S(\rho)=\sum_j\langle \omega_j|\rho|\omega_j\rangle$. I will drop the subscripts S and I and use the labels $\omega$ and $v$ to specify the ...
- 4,569
1
vote
Accepted
Two-photon N00N state through Mach-Zehnder interferometer
There are some corrections to the calculations, but it is more important to focus on the motivation: why are you sending a NOON state through a Mach-Zehnder interferometer? The idea of the NOON state ...
- 4,569
1
vote
Nielsen and Chuang: Solving equation of motion for amplitude damping
Think about how the density matrix is represented with the Bloch vector:
$$\tilde{\rho} = \frac{1}{2}(I + \vec{n} \cdot \vec{\sigma})$$
Then $$\frac{\mathrm{d}\tilde{\rho}}{\mathrm{d}t} = \dot{\vec{n}}...
- 21
1
vote
Diagonal elements of the transition dipole moment
The diagonal elements of the Hamiltonian give you information about the energy of a certain configuration. Not about the transitions.
For example, let's consider the Pairing Hamiltonian used in the ...
- 232
1
vote
Accepted
Does the entanglement have the conventional meaning in optical interferometers?
Entanglement is entanglement, there's just one definition of it (well, there's more complex variations like genuine multipartite entanglement etc, but that's not relevant here). However, (bipartite) ...
glS♦
- 26.4k
1
vote
What are the measurement operators $F_k$ corresponding to a homodyne measurement?
One thing that I feel is worth noting is that there are two ways to understand what "measurement" means in this context.
On the one hand, you have "measurements" in the sense of ...
glS♦
- 26.4k
1
vote
Accepted
Why do we use 'modes' in quantum optics?
Modes are governed by eigenfunctions, I agree. In quantum optics, we need more than just eigenfunctions to describe a state of light: we need to know how many photons have properties corresponding to ...
- 4,569
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