Questions tagged [gaussian-states]
10 questions
6
votes
1 answer
Trace of projected Gaussian operator product: puzzling difference between projection onto filled or empty fermionic mode
This question was motivated by a Mathoverflow posting.
I am comparing two traces of fermionic creation/annihilation operators $a_n,a_n^\dagger$ ($n=1,2,\ldots N$):
$$T_{\rm empty}= \operatorname{tr}\bigl(a_1 a_1^\dagger e^{a^\dagger X_1 a}a_1…
Carlo Beenakker
- 151
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6
votes
1 answer
Why is a state Gaussian if and only if its covariance matrix satisfies $\boldsymbol\sigma+i\boldsymbol\Omega\ge0$?
Let $\rho$ be a Gaussian state, described by the $2N\times 2N$ covariance matrix $\newcommand{\bs}[1]{{\boldsymbol{#1}}}\bs\sigma$. Denote with $\bs\Omega$ the $N$-mode symplectic form associated with the space:
$$\bs\Omega\equiv…
glS
- 27,510
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- 125
5
votes
1 answer
Finding the ground state of a Gaussian Hamiltonian
It is well known (arXiv) in continuous variable (CV) quantum information processing that an $N$-mode CV system whose dynamics are entirely Gaussian can be simulated efficiently with a classical computer as the dynamics can be simulated by computing…
atman
- 169
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4
votes
0 answers
Kraus operator sum rule for a convex-Gaussian channel
Consider a Gaussian operator$^{1}$ $M=Ce^{i\sum_{n,m=1}^N a^\dagger_n L_{nm} a_m}\equiv e^{ia^\dagger La}$, with $a_n^\dagger,a_n$ fermionic creation and annihilation operators, $C\in\mathbb{C}$, and $L$ a complex $N\times N$ matrix. I am not…
Carlo Beenakker
- 151
- 8
4
votes
1 answer
Does the covariance matrix of $\rho_1-\rho_2$ have a simple expression in terms of the individual covariance matrices?
Suppose $\rho_1$ and $\rho_2$ are two Gaussian states. The trace distance of a Gaussian state with covariance matrix $V$ can be computed as the sum of the eigenvalues of $i\Omega V$, so to find the distance of two Gaussian states we need the…
Noobgrammer
- 127
- 4
2
votes
1 answer
How to write the covariance matrix of a quantum gaussian state as a function of photon numbers?
Assume having a one-mode quantum Gaussian state with quadrature observable vector $\hat r = [\hat q , \hat p ] $ and covariance matrix $\sigma$. According to definition [1]:
\begin{equation}
\sigma = \text{tr}\left(\begin{bmatrix} \hat q^2 &…
hafezmg48
- 65
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1
vote
1 answer
Extending pairwise beam splitters operators to an overall transformation
Let $\bar x_s, \sigma_s$ be the vector of first moments and the covariance matrix of an $N$-mode Gaussian state $\rho_s$. If $\rho_s$ is paired with an empty environment of similar dimension, the total state is described by the…
Phil K.
- 13
- 3
1
vote
0 answers
Computing the QFI for a general Gaussian state
I found somewhere the following formula for the quantum Fisher information of a multimode Gaussian state $(\mathbf d_\theta, \sigma_\theta)$:
$$\tag{1}
H(\theta) = \frac{1}{2} \text{Tr}\left[ \left( \sigma_\theta^{-1} \partial_\theta \sigma_\theta…
Bentanglement
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1
vote
1 answer
Difference between Cluster states and Gaussian Boson Sampling- Discussion
Let me share my understanding first, the two key terms are Gaussian Boson Sampling and Cluster State. So far from my understanding, Gaussian Boson Sampling is a resonant state generator. I can generate this by sending a quantum emission ( I believe…
SakibulIslamSazzad
- 29
- 6
0
votes
0 answers
Writing pairwise beam splitters operators in the correct basis
Let $\bar x_s, \sigma_s$ be the vector of first moments and the covariance matrix of an $N$-mode Gaussian state $\rho_s$. If $\rho_s$ is paired with an empty environment of similar dimension, the total state is described by the…
Phil K.
- 13
- 3