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Questions tagged [quasiprobability-distributions]

For questions about quasiprobability distributions in quantum mechanics

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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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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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Can we get quasiprobability distributions other than $P,Q,W$ from generalised characteristic functions?

It's a standard result that the three well-known quasiprobability distributions can all be expressed in terms of the "$s$-ordered characteristic functions" as $$ W(\alpha) = \int\frac{d^2\...
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What is the observable corresponding to the $P$ function?

The P-function of a state $\rho$ (focusing on the single-mode case) can be written as, using a notation analogous to the one in Gerry&Knight' book, $$P_\rho(\alpha) = \int \frac{d^2\eta}{\pi^2} \...
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What is a "reciprocal ordering" when discussing quasiprobability distributions?

The Wikipedia page about the optical equivalence theorem mentions that, if we denote with $f_{\Omega}(\hat a,\hat a^\dagger)$ an "operator that is expressible as a power series in the creation ...
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Quasi-distribution for the composite (fermions + bosons) system

Let us have a system which consists of $N$ electrons with the spin (i. e. fermionic subspace has a dimension of $4^N$) and $K$ bosonic modes (let us consider $K=1$ for simplicity). Let us say, we ...
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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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Phase distribution of coherent states

I am studying the phase distribution for coherent states, as is defined in quantum optics. (See, for example, Introductory Quantum Optics by Gerry and Knight, pages 46–48). In this situation, we seek ...
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Distributions "more singular than a Dirac delta" must have negativity

I am looking at properties of the Glauber P-functions, which are distributions (in the sense of a dirac delta) on the complex plane, normalized so that $\int_{\mathbb{C}} d^2 \alpha P(\alpha) = 1$. On ...
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Is it possible to relate the expectation values of an operator w.r.t. two different density matrices if the matrices are related up to a displacement?

I'm trying to derive a general relationship between the expectation values of two different operators with respect to some unspecified state $E_{j}(\lambda)=\mathrm{Tr}[\rho \hat{O}_{j}(\lambda)]$ (...
quantum_loser's user avatar
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Do we actually need negative probabilities in quantum mechanics?

I was reading this thread and I'm a bit confused. The answer says negative probabilities can account for destructive wave interference and the events cancelling out. But if events just cancel out, ...
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Is there a way to find the superposition of two Wigner functions without finding their density operators?

Say I have two Wigner functions $W_1(x,p)$ and $W_2(x,p)$ representing pure states on optical phase space, and I want to know what the Wigner function of their superposition $W_{1+2}(x,p)$ looks like (...
Andrew Forbes's user avatar
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Intuitively, why does Quantum Mechanics involve a sum over all possibilities?

I understand that one can just mathematically derive the path integral from the Schrodinger equation. I'm looking for an intuitive explanation in contrast with classical mechanics. Consider a ...
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Is there a relationship between the phase space path integral and phase space quantum mechanics?

I understand that they're, in the end, related because they're the same theory. But is there a closer relationship because both are theories of probability distributions on phase space? I also ...
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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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Non-uniqueness of Glauber-Sudarshan $P$-function

For a state $\rho$ acting on single bosonic mode with coherent states $|\alpha\rangle$, one can always define a $P$-function to furnish a diagonal representation of the state in the coherent-state ...
Quantum Mechanic's user avatar
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Question about Glauber-Sudarshan $P$ representation

I'm reading Scully's 'Quantum Optics'. I've got some question about the Glauber-Sudarshan $P$ representation. It's straight forward that $$ P(\alpha) = \frac{e^{\vert \alpha \vert ^2}}{\pi^2} \int \...
Dylan_Wu's user avatar
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How can I show that the negative region of Wigner quasi-probability distribution is small enough than $h$?

Wigner quasi-probability distribution is the main tools used in the formulation of Quantum mechanics on phase space which is equivalent to usual formulations. This distribution can have negative ...
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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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Wigner function in relation to real space

The Wigner function is a quasi-probability distribution because it can have values of $x$ from -1,0,+1. As it is orthogonal to the $x,p$ plane can this be represented as a dipole moment along the $z$ ...
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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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Computing $\frac{1}{\pi^{2}} \int_{\mathbb{C}} \exp(\lambda\alpha^{*}-\lambda^{*}\alpha)\exp(-|\lambda|^{2}/2)d^{2}\lambda$

This came up when attempting to do a routine calculation of Wigner function of the vacuum state $$ \frac{1}{\pi^{2}} \int_{\mathbb{C}} \exp(\lambda\alpha^{*}-\lambda^{*}\alpha-|\lambda|^{2}/2)d^{2}\...
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Can the lifetime of a broken covalent bond be calculated?

According to a paper (molecular dynamic simulation) published in 2018,(https://www.scirp.org/journal/paperinformation.aspx?paperid=83351) the Maxwell-Boltzman distribution is also valid for the number ...
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What does it mean for $P$ functions to be "more singular than a delta"?

Consider the Glauber-Sudarshan $P$ representation of a state $\rho$, which is the function $\mathbb C\ni\alpha\mapsto P_\rho(\alpha)\in\mathbb R$ such that $$\rho = \int d^2\alpha \, P_\rho(\alpha) |\...
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15 votes
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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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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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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 ...
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