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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 Wigner function can become negative in certain region of phase-space. Some people claim that the negativity of this quasi-probability distribution signifies that the system behaves quantum mechanically (as opposed to classical physics, when probabilities are always positive). Apparently, there are still some controversies about this point. Please read the answers from the previously cited post: Negative probabilities in quantum physics

I would like to know whether there is an equivalence between the negativity of the Wigner distribution and some quantum behaviours or not. Is it still a question under debate / actual research or not ?

My main concern is that there are more and more experimental studies of the Wigner function (or other tomography captures) reporting negativity of the Wigner function. I would like to understand what did these studies actually probe.

As an extra question (that I could eventually switch to an other question): What is the quantum behaviour the negativity of the Wigner function may probe ?

Having not a lot of time at the moment, I would prefer an explicit answer rather than a bunch of (perhaps contradictory) papers regarding this subject. But I would satisfy myself with what you want to share of course :-)

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Let me split the "equivalence" in two parts:

Are there states with Wigner functions that are everywhere positive that show "quantum" behaviour?

The answer to this question is "yes". One very famous example are Gaussian (bosonic) states - their Wigner function, by definition, is a Gaussian (which is obviously positive) for an intro see e.g. Adesso et al. Nevertheless, they can be entangled and put into superpositions - the maximally entangled Bell state, for example, is in some sense a limit of Gaussian states. You can distill them (albeit not with the "Gaussian" operations, a restricted class of quantum operations, as shown by Giedke and Cirac). They can even (it seems, I haven't read the papers) violate Bell inequalities, see eg. Paternostro et al or Revzen et al. This should do as "quantum behaviour".

Hence positivity of the Wigner function does NOT imply that the state somehow behaves classically.

This leaves the other part of the question:

If a state has a Wigner function, which is negative at some point, does it show "quantum" behaviour?

I can't give a complete answer to this, as I don't know the literature well enough. However, for special states, this is possible. For example, it can be shown that $s$ waves (depending only on the hyperradius) are entangled iff their Wigner function is negative at some point, as seen in Dahl et al (once again, I've only skimmed the paper).

There is probably more (and I believe that there are probably people more inclined to foundations that know and work on these issues).

EDIT: There is more. I came across the topic today and found some very interesting papers that shed light on the other direction of the quantum state.

In fact, it was proven (Hudson 74) that the Wigner function of a pure quantum state is nonnegative if and only if the state is Gaussian. This answers the question sufficiently for pure states: Since there are entangled Gaussian states, there are states with nonnegative Wigner function that exhibit quantum behavior and as there are states that are separable, but not Gaussian (any product state consisting of non-Gaussian states I guess), there are states with negative Wigner function exhibiting no quantum behaviour.

The mixed-case seems to be still open, although you can find some progress here: Mandilara et al.

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  • $\begingroup$ Thanks a lot, I started to despair anyone would be of help. Thanks so much, I'll have a look on these papers as soon as possible. $\endgroup$ – FraSchelle Mar 14 '14 at 8:35
  • $\begingroup$ @FraSchelle I came across the topic again today and added the interesting links I found. It seems that positiveness of the Wigner function is a sign for Gaussianity rather than Quantumness $\endgroup$ – Martin May 12 '14 at 15:39
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There is recent work that answers this question in special contexts.

In particular, arXiv:1401.4174v1 establishes that for discrete systems of odd prime dimensions (i.e. we're talking about the discrete Wigner functions now, defined in arXiv:quant-ph/0401155v6 ), contextuality (a manifestly quantum property, see review by Mermin) is equivalent to negativity of the Wigner function.

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There is a paper Contextuality supplies the ‘magic’ for quantum computation telling you the equivalence between quantum contextuality, negativity of discrete wigner function, and the distillability of resource state for magic state distillation (for qubits) in the framework of fault-tolerant stabilizer computation.

If a qudit state can be used for magic state distillation (for qutrits see Hussain Anwar, Earl T. Campbell, and Dan E. Browne. Qutrit Magic State Distillation),the discrete wigner function of this state must contain negative items. Futhermore, this state must violate one of the non-contextuality hidden varibles inequalities as well.these inequalities are constructed from stabilizer measurements.

For qudits, these inequalities are state-dependent. That is to say, only a part of the state, which are distillable, can violate them. The equivalent relationship is quite simple---the amount of violation of the inequalities is proportional to the negativity of the phase point. However, not all phase point operators are used in the inequalities. In accordance to the resource theory of magic state distillation, there are only p*p facets are related.

For qubits, this construction fails to prove the equivalence between distillability and contextuality because of state-independence. Namely, every state, including the maximally mixed state, can violate the inequalities. But the negativity and contextuality are still equivalent.

For qubits, Robert Raussendorf etc. have done a remarkable work in the framework of measurement based quantum computation.(see. Contextuality as a resource for qubit quantum computation)

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