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 ...

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Understanding the relationship between Phase Space Distributions (Wigner vs Glauber-Sudarshan P vs Husimi Q)

I am moving into a new field and after thorough literature research need help appreciating what is out there. In the continuos variable formulation of optical state space. (Quantum mechanical/Optical)...
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Wigner Function for Thermal State

I am currently doing a reading on the subject of Quantum Optics from the book "Quantum Optics by Marlan O.Scully and M.Suhail Zubairy", I am currently learning about Quasi Probability distributions. ...
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What kind mathematical Transformation is this?

$$\int f(\alpha)e^{\alpha^*y-z|\alpha|^2}\pi^{-1}~\mathrm d^2\alpha~=~z^{-1}f(z^{-1}y)$$ I am doing the problem of finding Wigner function for fock states given by the exercise problem from ...
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How to Solve this Integral

I am currently doing a problem in Quantum optics, specifically the problem of finding Wigner Function for Number states or Fock states. I am actually did the problem in a different way and found that ...
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Physical Significance of Wigner Equation, Wigner Current

I am Currently doing a reading about Wigner function in Quantum optics, I learnt that the Wigner function maps the Quantum mechanical operators to phase space functions in phase space, so we can talk ...
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60 views

Ambiguity in True Quantum Phase-Space Distribution

In this paper, the following is stated: It is well known that the uncertainty principle makes the concept of phase space in quantum mechanics problematic. Because a particle cannot simultaneously ...
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137 views

Understanding the Mathematics of Wigner function [duplicate]

I fully understand that Wigner function provides the complete information of a state of a quantum system, i.e. quantum phase space, while not violating Uncertainty principle. But can anyone tell me ...
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59 views

How to transform a wigner function to represent loss of mode information (coarse graining)?

I have a highly multi-mode gaussian wigner function representing an optical field: $$W\left(\{p\},\{q\}\right)=\mathrm{Exp}\left(-\sum_{j=0}^{f}(b_{j}q^{2}_{j}+a_{j}p^{2}_{j})\right).$$ However the ...
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79 views

Wigner functions, symmetry

I'm trying to get more insight into quasiprobability distributions, as for example the Wigner function. There are some Wigner functions, which are symmetric. Symmetric: Fock state Thermal states ...
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82 views

Density matrix from Wigner distribution

Density matrix or Wigner function can be defined from the other with Fourier (or inverse) transformation. equivalently the value of W(q,p) can be seen as the mean value of the displaced parity ...
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54 views

Is there an equivalent of probability current for the Wigner distribution?

I know that for a wavefunction, I can derive a probability current $\mathbf{J}$ that satisfies the continuity equation: $$\nabla \cdot \mathbf{J}=-\frac{\partial}{\partial t} \big|\psi\big|^2$$ Can a ...
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217 views

What is the relation between phase space formulation with Wigner quasi-probability distributions and path integral formulation of quantum mechanics?

I am trying to conceptually connect the two formulations of quantum mechanics. The phase space formulation deals with Wigner quasi-probability distributions on the phase space and the path integral ...
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Necessary and sufficient conditions for a function to be the Wigner function of state

For any quantum state defined with a continuous position, the Wigner function is a quasiprobability distribution on phase space. It has many properties, such as that its marginal are probability ...
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57 views

Do non-Gaussian states always show negativity in phase space? [closed]

According to Hudson’s theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. In cases, in which the existing well-known Hudson theorem immediately tells that ...
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118 views

Are the Wigner and Husimi transforms injective?

I am wondering if the Wigner function is injective. By injective I mean, that, for every density matrix $\rho$, there is a different Wigner distribution. The same question applies to the Husimi ...
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188 views

Bopp operators and Wigner-Weyl representation

I am learning about the Wigner-Weyl transformations to move a $c$-number Lindblad operator $A(x,p)$ back into operator form. As far as I know, to move back and forth normally requires a four variable ...
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411 views

Good source for numerical simulations of Wigner function?

I'm interested in simulating the time evolution of a Wigner function for a harmonic oscillator (and possibly some other potentials) and I can't seem to find a good resource for that. My background in ...
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409 views

What is discrete phase space?

I've been reading a little about the usual, continuous Wigner functions and phase space quasi-distributions in general, and I believe I understand the idea behind them. The Wigner function arises when ...
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How to check whether a given $W(x,p)$ represents a Wigner function of a physical state? [duplicate]

For simplicity let us consider one-dimensional quantum-mechanical systems only. Given any state $\rho\in\mathcal{B}(\mathcal{H})$ and its Wigner function $W_\rho(x,p)$, there are several properties it ...
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113 views

How can I take the Wigner transform of an operator with an absolute value?

I want to be able to find the Wigner transforms of operators of the form $\Theta(\hat{O})$, where $\Theta$ is the Heaviside function and $\hat{O}$ in general depends on both $x$ and $p$. For the ...
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270 views

Examples of Weyl transforms of nontrivial operators

I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more ...
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131 views

Deriving probability distributions from the Wigner distribution

I know that I can calculate the probability distributions of $x$ and $p$ from the Wigner quasiprobability distribution, and I can calculate the probability distributions of other operators by ...
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Are they the same thing: Wigner distribution in quantum Boltzmann equation and Wigner function in quantum optics?

We know that quantum Boltzmann equation (QBE) is an equation of motion for the interacting Green's function $G^<(\vec{x}_1,t_1;\vec{x}_2,t_2)\equiv\mathrm{i}\langle \psi^\dagger(x_2)\psi(x_1)\...
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Are negativity of the Wigner function and quantum behaviour equivalent?

I've read the following question: Negative probabilities in quantum physics and I'm not sure I understand all the details about my actual question. I think mine is more direct. It is known that the ...
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243 views

Motivation for Wigner phase space distribution

Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula $$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\...
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What is the physical interpretation of the density matrix in a double continuous basis $|\alpha\rangle$, $|\beta\rangle$?

(a) Any textbook gives the interpretation of the density matrix in a single continuous basis $|\alpha\rangle$: The diagonal elements $\rho(\alpha, \alpha) = \langle \alpha |\hat{\rho}| \alpha \...
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Operator norm directly from phase space representation of photonic quantum operator

I'm interested in calculating the operator norm of a Hermitian operator, say $B$, acting on the Hilbert space of square integrable functions. The context is I have an optical system in all its ...
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Interpretation of Wigner function in optics

I work in the field of synchrotron radiation sources where radiation (often x-rays) is produced from an electron beam going through magnetic fields. The quality of the resulting x-ray beam is ...
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Negative probabilities in quantum physics

Negative probabilities are naturally found in the Wigner function (both the original one and its discrete variants), the Klein paradox (where it is an artifact of using a one-particle theory) and the ...
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an example of a quantum system for which wigner function transitions to negative values

I want to check my understanding of the Wigner transform and try to understand why and how exactly the probabilistic interpretation drops down as the function goes to zero and then to negative values ...