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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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
$$
\...
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Generating function for Weyl-ordered spin correlation functions
Consider the standard 2-dimensional phase space: $[\hat{x},\hat{p}] = \mathrm{i}$. Upon taking the Wigner transform of the density matrix $\hat{\rho}$,
\begin{equation}
W(x,p) = \int\mathrm{d}y\, \...
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Looking at the Wigner function plot of a given state, how do we know if it's a good option for use as a qubit?
I am looking at GKP and cat states. Wigner plots are newish to me, and I'm trying to get an intuitive feel for the imformation they impart.
Just by looking at a Wigner plot, can one discern whether ...
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Fourier Transform of $s$-ordered Characteristic Function
In the book, "Quantum Continuous Variables (A Primer of Theoretical Methods)" by Alessio Serafini, on page 70, he defines an $s$-ordered characteristic function to be:
$$
\chi_s(\alpha)=\...
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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: $$
\hat{D}(\mathbf{r})=\exp \...
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Mismatch in quantum state purities
Method 1:
(i) Expand the density operator in the Fock basis as
$\rho=\sum_{m,n}{\rho_{mn}|m\rangle\langle n|}$.
(ii) Purity = $tr{\rho^2}=\sum_{m,n,f,g, \lambda}{\rho_{mn}\rho_{fg}\langle \lambda|m\...
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Classical limit of Moyal bracket in integral representation
It is well-known that the Poisson bracket can be recovered out of the Moyal bracket under the limit when $\hbar$ goes to zero $$\lim_{\hbar\rightarrow 0}\frac{1}{i\hbar} \lbrace f,g\rbrace_M=\lbrace f,...
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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
$$\tilde Φ_σ(α) = \frac{1}{π σ^2} e^{-...
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Is there a canonical Taylor expansion for operators in terms of $X$ and $P$?
Consider the algebra of operators acting on wavefunctions ($L^2(\mathbb{R})$) generated by $X$ and $P = -i\hbar (\partial/\partial x)$. For some operator $A$ in this algebra, or possibly in a ...
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Boundary conditions for Jordan-Wigner transformation for spin model consistency
For a transverse field ising model we can use a Jordan-Wigner transformation to solve the system. Following "The Quantum Ising chain for beginners" we have $\sigma_j^z = 1-2c_j^\dagger c_j$ ...
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An equation for Wigner's quasi-probability distribution?
I learned that Moyal's evolution equation is the equation for the time-evolution of Wigner's quasi-probability distribution. However, I couldn't say I perfectly understand the meaning of this ...
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Wigner transform, convolution, and poles
Let
\begin{equation}
\int\mathrm{d}z~ A(x,z) B(z,y) = \delta(x - y).
\end{equation}
Taking Wigner transform of both sides we readily obtain
\begin{equation}
A^W(X,p) \star B^W(X,p) = 1,
\end{equation}
...
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How do you get the distribution of a linear combination of the quadratures using the Wigner function?
I know you can get the distribution of a quadrature, x or p, by integrating the Wigner function over all the rest of variables (2N-1 variables for an N mode state). But how do you get the ...
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What is the most general wave function of a minimum uncertainty (Gaussian) state in quantum mechanics?
For some state $|\psi\rangle$ it is possible to recover the uncertainty principle using the fact that $$\left|(\hat{\sigma_{Q}}-i\lambda\hat{\sigma_{P}})|\psi\rangle\right|^{2}\geq0,$$where$$\hat{\...
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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 ...
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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
$$ [A(\textbf{r})B(\textbf{r})]_{w}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{4}}e^{i\...
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Anyone got any book recommendations for the derivation of the Wigner formulation of quantum mechanics?
Preferably one that starts from classical mechanics and derives the theory from that
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Trouble proving Wigner function identity [closed]
I am trying to prove $$\int d^2 \alpha W(\alpha)=1$$ where $W(\alpha)$ represents the Wigner funcion. However, I have trouble solving it. I tried solving it as follows but I think I have done some ...
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What does $\langle -z|$ mean in this expression $\langle -z|\rho|z\rangle$?
In some formulas of the Wigner function, there is an expression $\langle -z|\rho|z\rangle$ where $|z\rangle$ represents a coherent state. I don't understand what $\langle - z|$ means. Some other ...
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Is there a simple way to obtain the Quantum Fisher Information (QFI) from a Wigner function? (or any other quasiprobability distribution?)
In theory the Wigner function $W(q,p)$ contains all the same information as the density matrix $\rho$ associated with it, so it's definitely possible to write the QFI $\mathcal F_Q$ in terms of the ...
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Are Wigner functions of any Unitary operator in $B(L^{2}(\mathbb{R}))$ in $L^{2}(\mathbb{R}^{2})$?
Are Wigner functions of any Unitary operator in $B(L^{2}(\mathbb{R}))$ in $L^{\infty}(\mathbb{R}^{2})$?
i.e.
Let $e^{iA} \in B(L^{2}(\mathbb{R}))$. Define the Wigner function (Wigner transform) as ...
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Momentum probability density from Wigner distribution
I want to prove that $|\hat{\psi}(p)|^2= \frac{1}{2\pi} \int W_\psi \mathrm{d}x $ where $W_\psi $ is the Wigner function.
Starting with the definition I get ($z=-y$ and $u=x+z/2$):
$$\frac{1}{2\pi}\...
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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 ...
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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 ...
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Reconciling two descriptions of the Wigner function
A classical way to define the Wigner function ($\hbar=2$) of a density operator $\rho$ is as follows for $x=(x_{1}, x_{2})^{T}$:
$$W(x) = \frac{1}{4\pi} \int^{\infty}_{-\infty} d\xi \exp(\frac{-i}{2}...
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Jordan-Wigner transformation particle exchnage
Suppose you have a spinless fermionic system:
$\hat H = -t \displaystyle\sum_i (c^+_ic_{i+1} + h.c.)$
The Jordan Wigner transformation acts as:
$c^+_i = Z_0\otimes..\otimes Z_{i-1} \otimes \sigma^+\...
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Does the integral of a Wigner function over a finite region mean anything?
I've recently been dipping my toes deeper into the so-called "Wigner function" formalism for quantum theory, and what I am curious about is this: ostensibly, the Wigner function is the ...
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How can the Wigner function of squeezed states be non-negative?
It is always said that when the Wigner function of quantum states takes a negative value, then it is a clear signature of non-classicality of this particular state. It is also well-known that the ...
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Wigner Transform in momentum space coefficient
I am currently reading some about the Wigner transform, and I ran into a problem. The Wigner transform (in the literature I am reading) is defined as:
$$\tilde{A} = \int \exp{\big[\frac{-ipy}{\hbar}\...
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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 ...
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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 ...
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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 ...
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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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Formulations of the Wigner function in Quantum Field Theory (QFT in Phase Space)
I'm studying the phase space formulation of quantum field theory for my final degree project, and I have found two very different ways to construct the Wigner funtion. In the first method, a phase ...
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Multiplicative inverse of Weyl symbol and invertibility of operator
If the Weyl symbol $A_W$ of an operator $\hat{A}$ has a multiplicative inverse at every point of the phase-space, can I conclude that $\hat{A}$ is invertible?
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Wigner transform of a thermal density matrix (Schrödinger Hamiltonian)
I am interested in computing the Wigner transform in $R^{2n}$ of $$e^{-\beta H}$$
where
$$H = \sum_{k=1}^n P_k^2 + V(x_1, \ldots,x_n)$$
I am assuming that $V$ is a polynomial bounded below and ...
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Wigner transform & convolution
I'm trying to understand the gradient expansion within the Keldysh formalism. In particular, I am reading "Quantum Field Theory of Non-equilibrium States" by J. Rammer, section 7.2, ...
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General quantum operator
Is it true that any operator can be expressed as
(e.g. in one dimension)
$$\hat{A}=\sum_{n=0, \, m=0}^{\infty}c_{n,m}\hat{x}^n\hat{p}^m \, ?$$
It seems true because any classical observable is a ...
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Wigner-Weyl transform for a function of coordinates only
I am reading this paper by Tatarskii, which serves as an introduction to the Wigner representation of quantum mechanics.
There is a step in the paper involving the Weyl transform that does not seem ...
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Change of variables for momenta [closed]
http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stud_seminar/Wigner_function.pdf
From the Appendix in the above PDF (page 945), below equation (A3) the following expressions are given:
$$
u = ...
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Is the Wigner function a signed measure?
I have read in Wikipedia that quasiprobability distributions in phase space quantum mechanics may fail to be $\sigma$-additive, but I don't know in which sense this is true. If I have a Wigner ...
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How does the Wigner function differ from quantum distribution functions?
I understand that Wigner functions are quantum-mechanical phase-space distribution functions (quasi-distribution to be more specific). For spin-1/2 particles the Wigner function is a $4\times 4$ ...
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Prove that $f_\psi(x,p)$ is the Wigner Function of a pure state iff $H\star f_\psi= E f_\psi$
Given a pure state $|\psi\rangle$ with position wavefunction $x\mapsto\psi(x)$, define its Wigner function as
$$f_\psi(x,p) = \frac{1}{2\pi} \int dy e^{-iyp} \psi(x+y/2)\psi^*(x-y/2)
\equiv \frac{1}{2\...
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Wigner Function and Spin in the Classical Limit?
This is something I got curious about. Let's say I have the Wigner function for an $n$ particle system:
$$W \equiv W(x_1,\dots,x_n,;p_1,\dots,p_n) $$
Now, let's say this system obeys has spin. As far ...
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Generalization of Wigner overlap formula
I want to generalize the Wigner overlap formula,
$Tr( F G ) = 2 \pi \int_{-\infty}^{\infty} dq \int_{-\infty}^{\infty} dq W_F(q,p) W_G(q,p)$,
where $W_F(q,p)$ and $W_G(q,p)$ are the Wigner functions ...
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