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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 $\rho$...
Nicolas Medina Sanchez's user avatar
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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\...
Landuros's user avatar
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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 ...
compp's user avatar
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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 ...
Song of Physics's user avatar
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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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Quantum mechanics representations $x$, $p$, and not $(x,p)$? [duplicate]

In Quantum Mechanics, generally speaking we work with space representation $\Psi(x)$ and/or the momentum representation $\Psi(p)$ of wavefunctions. Are there representations with mixing of $(x,p)$, i....
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How does the Wigner function transform when the transformation is provided in $(x,p)$-form?

In quantum optics, when a transformation of canonical variables $\hat{x},\hat{p}$ is provided, how does the Wigner function change under the transformation? For example, when the transformation is $$ \...
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Generating function for Weyl-ordered spin correlation functions

Consider the standard 2-dimensional phase space: $[\hat{x},\hat{p}] = \mathrm{i}$. Upon taking the Wigner transform of the density matrix $\hat{\rho}$, \begin{equation} W(x,p) = \int\mathrm{d}y\, \...
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Fourier Transform of $s$-ordered Characteristic Function

In the book, "Quantum Continuous Variables (A Primer of Theoretical Methods)" by Alessio Serafini, on page 70, he defines an $s$-ordered characteristic function to be: $$ \chi_s(\alpha)=\...
Pratham Hullamballi's user avatar
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Discrepency regarding term in the Wigner Characteristic Function

Consider the quantum characteristic function, $$ \chi_\rho(\mathbf{s})=\operatorname{tr}\{\hat{\rho} \hat{D}(\mathbf{s})\} $$ The displacement operator is defined to be: $$ \hat{D}(\mathbf{r})=\exp \...
Pratham Hullamballi's user avatar
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Mismatch in quantum state purities

Method 1: (i) Expand the density operator in the Fock basis as $\rho=\sum_{m,n}{\rho_{mn}|m\rangle\langle n|}$. (ii) Purity = $tr{\rho^2}=\sum_{m,n,f,g, \lambda}{\rho_{mn}\rho_{fg}\langle \lambda|m\...
Saurabh Shringarpure's user avatar
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Classical limit of Moyal bracket in integral representation

It is well-known that the Poisson bracket can be recovered out of the Moyal bracket under the limit when $\hbar$ goes to zero $$\lim_{\hbar\rightarrow 0} \lbrace f,g\rbrace_M=\lbrace f,g \rbrace_P.$$ ...
Nicolas Medina Sanchez's user avatar
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What is the Weyl transform of narrow Gaussians and/or the Dirac delta?

Consider the family of Gaussians in $q$, $p$ with decreasing widths $σ$ $$Φ_σ(q,p) = \frac{2}{π σ^2} e^{-\frac{2}{σ^2}(q^2+p^2)}$$ or in complex plane coordinates $$\tilde Φ_σ(α) = \frac{1}{π σ^2} e^{-...
The Vee's user avatar
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Is there a canonical Taylor expansion for operators in terms of $X$ and $P$?

Consider the algebra of operators acting on wavefunctions ($L^2(\mathbb{R})$) generated by $X$ and $P = -i\hbar (\partial/\partial x)$. For some operator $A$ in this algebra, or possibly in a ...
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An equation for Wigner's quasi-probability distribution?

I learned that Moyal's evolution equation is the equation for the time-evolution of Wigner's quasi-probability distribution. However, I couldn't say I perfectly understand the meaning of this ...
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Wigner transform, convolution, and poles

Let \begin{equation} \int\mathrm{d}z~ A(x,z) B(z,y) = \delta(x - y). \end{equation} Taking Wigner transform of both sides we readily obtain \begin{equation} A^W(X,p) \star B^W(X,p) = 1, \end{equation} ...
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How do you get the distribution of a linear combination of the quadratures using the Wigner function?

I know you can get the distribution of a quadrature, x or p, by integrating the Wigner function over all the rest of variables (2N-1 variables for an N mode state). But how do you get the ...
eternalstudent's user avatar
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What is the most general wave function of a minimum uncertainty (Gaussian) state in quantum mechanics?

For some state $|\psi\rangle$ it is possible to recover the uncertainty principle using the fact that $$\left|(\hat{\sigma_{Q}}-i\lambda\hat{\sigma_{P}})|\psi\rangle\right|^{2}\geq0,$$where$$\hat{\...
Adrien Amour's user avatar
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Do we lose information about the state when we obtain the Wigner function by solving the eigenvalue equation?

It can be shown that $$H(q,p)\star W_{\psi}(q,p)=EW_{\psi}(q,p)$$ where $H(q,p)$ is the classicaly Hamiltonian function, $\star$ is the Moyal/Groenewold star product and $W_{\psi}(q,p)$ is the Wigner ...
Adrien Amour's user avatar
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Obtaining the star product from the Weyl quantisation of the product of two symbols

It can be shown (Groenewold 1946) that the Weyl quantisation of the product of two Weyl symbols is given by $$ [A(\textbf{r})B(\textbf{r})]_{w}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{4}}e^{i\...
Adrien Amour's user avatar
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Anyone got any book recommendations for the derivation of the Wigner formulation of quantum mechanics?

Preferably one that starts from classical mechanics and derives the theory from that
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Trouble proving Wigner function identity [closed]

I am trying to prove $$\int d^2 \alpha W(\alpha)=1$$ where $W(\alpha)$ represents the Wigner funcion. However, I have trouble solving it. I tried solving it as follows but I think I have done some ...
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What does $\langle -z|$ mean in this expression $\langle -z|\rho|z\rangle$?

In some formulas of the Wigner function, there is an expression $\langle -z|\rho|z\rangle$ where $|z\rangle$ represents a coherent state. I don't understand what $\langle - z|$ means. Some other ...
Anaya's user avatar
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Is there a simple way to obtain the Quantum Fisher Information (QFI) from a Wigner function? (or any other quasiprobability distribution?)

In theory the Wigner function $W(q,p)$ contains all the same information as the density matrix $\rho$ associated with it, so it's definitely possible to write the QFI $\mathcal F_Q$ in terms of the ...
Andrew Forbes's user avatar
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Are Wigner functions of any Unitary operator in $B(L^{2}(\mathbb{R}))$ in $L^{2}(\mathbb{R}^{2})$?

Are Wigner functions of any Unitary operator in $B(L^{2}(\mathbb{R}))$ in $L^{\infty}(\mathbb{R}^{2})$? i.e. Let $e^{iA} \in B(L^{2}(\mathbb{R}))$. Define the Wigner function (Wigner transform) as ...
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Momentum probability density from Wigner distribution

I want to prove that $|\hat{\psi}(p)|^2= \frac{1}{2\pi} \int W_\psi \mathrm{d}x $ where $W_\psi $ is the Wigner function. Starting with the definition I get ($z=-y$ and $u=x+z/2$): $$\frac{1}{2\pi}\...
Silas's user avatar
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Derivation of Wigner Function of cosine with phase

The Wigner Function of $x(t)$ is $$ W(t,f) = \frac{1}{2\pi}\int x\Big(t+\frac{\tau}{2}\Big)x^*\Big(t-\frac{\tau}{2}\Big) e^{-j2\pi f\tau}\;d\tau $$ I know how to get the $W(t,f)$ of $$x(t)=\cos(2\pi ...
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Converting the complex Wigner function to its real form in terms of the quadrature operators

I noticed something that bugged me recently, the Wigner function which is defined for one mode in the complex plane as $$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{...
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Construct the Density Matrix of a Gaussian State from its First and Second Moments / Wigner Function

Borrowing some description for the setup from a question I posted earlier here; Suppose we have $N$ bosonic modes (or quantum harmonic ocsillators) with the usual commutation relations. Now define the ...
Lost In Euclids 5th Postulate's user avatar
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What is the Wigner representation of $\left(\hat{x}^2+\hat{p}^2\right)^n$?

I would like to calculate the Wigner representation of the operators $\left(\hat{x}^2+\hat{p}^2\right)^3$ and $\left(\hat{x}^2+\hat{p}^2\right)^4$. I know at least two ways to do it, but both rely on ...
Marcos Gil's user avatar
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2 answers
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Reconciling the expression for the Wigner function involving $\langle x+\xi/2|\rho|x-\xi/2\rangle$ with the one using the characteristic function

A classical way to define the Wigner function ($\hbar=2$) of a density operator $\rho$ is as follows for $x=(x_{1}, x_{2})^{T}$: $$W(x) = \frac{1}{4\pi} \int^{\infty}_{-\infty} d\xi \exp(\frac{-i}{2}...
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Jordan-Wigner transformation particle exchnage

Suppose you have a spinless fermionic system: $\hat H = -t \displaystyle\sum_i (c^+_ic_{i+1} + h.c.)$ The Jordan Wigner transformation acts as: $c^+_i = Z_0\otimes..\otimes Z_{i-1} \otimes \sigma^+\...
relaxon's user avatar
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Does the integral of a Wigner function over a finite region mean anything?

I've recently been dipping my toes deeper into the so-called "Wigner function" formalism for quantum theory, and what I am curious about is this: ostensibly, the Wigner function is the ...
The_Sympathizer's user avatar
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How can the Wigner function of squeezed states be non-negative?

It is always said that when the Wigner function of quantum states takes a negative value, then it is a clear signature of non-classicality of this particular state. It is also well-known that the ...
FlyGuy's user avatar
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Wigner Transform in momentum space coefficient

I am currently reading some about the Wigner transform, and I ran into a problem. The Wigner transform (in the literature I am reading) is defined as: $$\tilde{A} = \int \exp{\big[\frac{-ipy}{\hbar}\...
user132849's user avatar
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1 answer
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Calculating the Wigner transform of operators

Recently I started to study the formulation of quantum mechanics in the phase space. So I was introduced to the concept of Wigner function and Weyl transform. I learned that if F is an operator, then ...
Wagner Coelho's user avatar
2 votes
1 answer
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Understanding derivation of Wigner function for the Harmonic oscillator

In the document https://www.hep.anl.gov/czachos/aaa.pdf, they derive the Wigner functions, $f_n$ for the harmonic oscillator. However, I have some problems understanding some of the steps. On page 37 ...
Logi's user avatar
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Effects of non-locality in the star-product of two fields

My question regards an argument appearing on page 19 of the review: Quantum Field Theory on Non-commutative Spaces - Szabo. The Fourier integral kernel representation of the star-product of two fields ...
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How is it possible to find the Wigner function for spin coherent states?

I studied Wigner function distribution for Glauber coherent state and I know that by using this function we can find the probability distribution for particle's position, but How can me find and ...
Aisha's user avatar
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Wigner functional for fermionic fields (QFT in phase space)

I'm curently studying the Wigner functional formulation of Quantum Field Theory, which is derived from the Schrödinger picture: the operators which act on the states of the Fock space are functions of ...
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