Questions tagged [wigner-transform]
The Wigner transform is the bridge between Hilbert space operators to phase-space quantities (c-numbers). Use for issues relating to the Weyl correspondence (the inverse of the Wigner transform), the Wigner function (the Wigner-transform of the density matrix) and, in general, Quantum Mechanics in phase space issues, such as the *-product, the Wigner transform of the operator multiplication operation. May also use for distributions such as the Husimi.
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What does the superposition of fields mean in the context of the convolution of two Glauber-Sudarshan $P$-representations?
In his 1963 paper, in which he introduces his formulation of the Glauber-Sudarshan $P$-representation (https://doi.org/10.1103/PhysRev.131.2766), Glauber refers to the convolution of the $P$-...
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Critical Points of a Wigner function
I am interested in calculating the critical points of a Wigner function
$$
W(x,p)=\frac{1}{\pi}\int_{-\infty}^\infty\left\langle x+y\middle|\rho\middle|x-y\right\rangle e^{-2ipy}\mathrm{d}y
$$
...
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Is there a non-negative normalized Wigner function that doesn't correspond to a physical state?
This is related to Is the Wigner function non-negative only for convex mixtures of Gaussian states? and Can the characteristic function $\chi_\rho(\beta)={\rm tr}[\rho D(\beta)]$ be an indicator ...
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Is the Wigner function non-negative only for convex mixtures of Gaussian states?
Hudson's theorem, the result usually cited in this context, tells us that for a pure state, the Wigner is non-negative iff the state is Gaussian, but doesn't in general say anything about mixed states....
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Can the characteristic function $\chi_\rho(\beta)={\rm tr}[\rho D(\beta)]$ be an indicator function?
Given the characteristic function defined as:
$$\chi(\beta)=\text{tr}[\rho D(\beta)],$$
with $D(\alpha)=e^{\alpha a^\dagger-\bar\alpha a}$ the displacement operator. Is it possible that for some $\rho$...
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How is Hudson's theorem for the Wigner function proved?
Hudson's theorem tells us that a pure state has non-negative Wigner function iff it's Gaussian. This was originally proven in [Hudson 1974], and then generalised to multidimensional systems in [Soto ...
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Dirac delta of operators multiplying matrix element
In playing around with the Wigner-Weyl correspondence, I found myself needing to perform an integral of exponential operators, which I am confused about. TLDR: help to evaluate $$\int d{x}dy\ \delta(x\...
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Is there a probability distribution associated with fermionic Gaussian states
I am writing this as a mathematician trying to understand fermionic Gaussian states.
Up to global phase, a quantum state can be faithfully represented in terms of a quasi-probability distribution on ...
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How does squeezed multimode light, where the modes are entangled, behave in a beamsplitter?
I understand how to work with and describe squeezed single modes going through a beamsplitter, and can conceptually talk about what's happening. If I now take a source of squeezed light that has ...
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Wigner transform of $O_1 O_2$ in terms of Wigner transforms of $O_1$ and $O_2$?
The Wigner-Weyl transform of a quantum operator $O$ is defined as
$$
W[O](q,p) = 2 \int_{-\infty}^{\infty} dy\ e^{- 2 i p y} \langle q + y | O | q - y \rangle \ dy
$$
and then given a density matrix $\...
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Quantum Behavior and Negativity of Wigner Functions
Let us consider a scenario where we have a dataset $\mathbf{X}$, which is a collection of vectors $\mathbf{x}_i \in \mathbb{R}^n$. We encode each component $x_j \in \mathbb{R}$ of $\mathbf{x}$ in a ...
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The Lorentz-non-covariance of the Wigner Function
What does the fact that the Wigner function is not Lorentz-covariant imply?
My analysis so far led me to the (probably naive) understanding that there really is nothing special about it, just that it ...
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Marginal of Wigner Function calculation
I am reading on the Wigner function from the Gerry and Knight book. It defines it as:
$$ W(q, p) \equiv \frac{1}{2 \pi \hbar} \int_{-\infty}^{\infty}\left\langle q+\frac{1}{2} x|\hat{\rho}| q-\frac{1}{...
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Wigner function of $|n\rangle\langle m|$
For $n=m$, the Wigner function is given by,
$$
W_n(\alpha) = \frac{2}{\pi} (-1)^n \exp(-2 |\alpha|^2) L_n(4 |\alpha|^2),
$$
And for $n \neq m $, it is,
$$
f_{mn}=\sqrt{\frac{m!}{n!}} e^{i(m-n) \arctan\...
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How does the Stratonovich-Weyl operator kernel, used to find the Wigner function, work?
Recently during my studies, I came across an alternative construction of the Wigner function. This construction starts from the notion of the Stratonovich-Weyl operator kernel. I saw this construction ...
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