• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

204 submissions , 162 unreviewed
5,026 questions , 2,180 unanswered
5,344 answers , 22,686 comments
1,470 users with positive rep
815 active unimported users
More ...

  How to check whether a given $W(x,p)$ represents a Wigner function of a physical state?

+ 7 like - 0 dislike

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 always satisfies,

  • it is real-valued and bounded in absolute value by $2/\hbar$ from above,
  • its marginals are well-behaved probability distributions,
  • its overlap $\int W_\rho(x,p) W_\sigma(x,p) \mathrm{d}x\mathrm{d}p$ with the Wigner function of any other state $\sigma$ is nonnegative.

Given an arbitrary function $W(x,p)$, we can reconstruct a state $\rho$ it corresponds to iff all of these conditions are met using the Weyl transform. If some properties are not satisfied, the resulting operator will be unphysical—non-Hermitian, unnormalized or non-positive. The question is how to check the conditions prior to the calculation.

The first two are usually fairly easy to check but the latter involves going through all states of the system. Even if we restrict ourselves to pure states, this still iterates over all rays in an infinite-dimensional linear vector space and is as such usually done finding a proof tailored to the specific case. Is there a simpler (algorithmic) condition which can replace the third bullet above and implies its validity?

This post imported from StackExchange Physics at 2014-09-02 20:42 (UCT), posted by SE-user Vašek Potoček
asked Sep 2, 2014 in Theoretical Physics by Vašek Potoček (45 points) [ no revision ]

This is very difficult to check explicitly, since in the Wigner description, quantum positivity is not a natural condition. One usually proceeds in the opposite direction: one knows that a state is positive, then calculates its Wigner transform, and can be sure that the required conditions are met. 

1 Answer

+ 5 like - 0 dislike

There is another formulation, not algorithmic, but the exact analog of positiv definiteness in Bochner's theorem for Fourier transforms of probability measure. This is called $\omega$-positivity (or $\omega$-positive definiteness, or $\omega$-non negative definiteness). The (inverse) Fourier transform of the Wigner function has to be $\omega$-positive, a condition like positive definiteness, but with an additional function called $\omega$, see the paper

Robin L. Hudson, Quantum Bochner theorems and incompatible observables,  Kybernetika (2010) Volume 46, Issue: 6, page 1061-1068, ISSN: 0023-5954, http://eudml.org/doc/196490

and the references therein.

answered Sep 3, 2014 by UwF (90 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights