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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 Wigner function of squeezed states is completely non-negative, yet, we still call squeezed states quantum. How does one explain this supposed contradiction here? is the Wigner function just not a "strong enough" measure of non-classicality? if so, what is? Also, what defines non-classicality for that matter (specifically for squeezed states)?

Thanks!

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  • $\begingroup$ I don't think squeezed states are supposed to be necessarily quantum, right? In the simplest 2 particles in 1d oscillator framework, don't they classically describe the two particles being a quarter period out of phase with each other? $\endgroup$
    – user1379857
    Commented Jun 26, 2021 at 1:11
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    $\begingroup$ @user1379857 squeezed states are non-classical. $\endgroup$
    – ZeroTheHero
    Commented Jun 26, 2021 at 1:35
  • $\begingroup$ Take the second order correlation function $g^2(0)=\dfrac{\langle a^\dagger a^\dagger a a \rangle}{\langle a^\dagger a \rangle ^2}$ in the standard Hanbury Brown Twiss experiment. Now, $g^2(0)<1$ surely gives you quantum states of light, but does $g^2(0)>1$ forbid you from using the quantum picture? I don't think so. It is just that classical methods work there, and you can choose not to talk about quantum mechanics. $\endgroup$
    – Physiker
    Commented Jun 26, 2021 at 8:48
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    $\begingroup$ Thanks guys. Truth is, now I am even more confused because you're basically confirming what I'm thinking. Correct me if I'm wrong, so there's no one "common"/"universal" operator/indicator for non-classicality. Essentially, any measure that "breaks" what is known classically is eligible to be such a candidate, but it doesn't mean that it'll work for every state that might be quantum... $\endgroup$
    – FlyGuy
    Commented Jun 26, 2021 at 10:50
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    $\begingroup$ @FlyGuy that’s basically correct. There is no universal witness for this, although the behaviour of singularity of the P-function is as close as you get. $\endgroup$
    – ZeroTheHero
    Commented Jun 26, 2021 at 13:16

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There is no contradiction because positivity of the Wigner is not enough to guarantee classicality: squeezed states are precisely examples of this. What is known for pure states (Hudson’s theorem) is that only Gaussian states have non-negative WFs (at least in the $xp$ plane), but the theorem is silent on the classicality of the Gaussian states.

Most non-classical states have singular $P$-functions, although it is recognized that the criteria is not so simple and there are recent results in

Damanet, F., Kübler, J., Martin, J. and Braun, D., 2018. Nonclassical states of light with a smooth P function. Physical Review A, 97(2), p.023832

that suggest a way of engineering non-classical states so that their P-function behaves more smoothly.

I do not know of a necessary and sufficient condition for a state to be classical, although negativity in the Wigner function guarantees it is NOT.

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  • $\begingroup$ Thanks for the answer ZeroTheHero. This is still kind of baffling me. So what criteria do people use to say that squeezed states are actual quantum states? $\endgroup$
    – FlyGuy
    Commented Jun 26, 2021 at 10:58
  • $\begingroup$ The link to the wiki page provided above will partially answer your question: squeezed states cannot be described in classical E&M. This isn’t a very satisfactory answer but I’m afraid that if you are looking for a mathematical criterion there just isn’t one yet. $\endgroup$
    – ZeroTheHero
    Commented Jun 26, 2021 at 13:14
  • $\begingroup$ to add to this: there are no known easily computable necessary and sufficient criteria for nonclassicality. There are, however, hierarchies/classes of criteria which are necessary and sufficient, in the sense that one is ensured that if a state is nonclassical, then one of the criteria wlil witness such nonclassicality (but of course, one can never know using only a finite number of these criteria that the state is classical; you can only gather increasing amounts of evidence towards that hypothesis). See e.g. journals.aps.org/prl/abstract/10.1103/PhysRevLett.89.283601 $\endgroup$
    – glS
    Commented Jun 26, 2021 at 15:20
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    $\begingroup$ @glS ah yes. I was looking for that reference but could not remember the authors or the details. Thanks for linking to this. $\endgroup$
    – ZeroTheHero
    Commented Jun 26, 2021 at 15:31
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    $\begingroup$ @ZeroTheHero I would also add that Hudson theorem, as far as I'm aware, is only valid for pure states. For pure states, non-negativity of the Wigner is equivalent to the Gaussianity of the state. I don't think the equivalence holds more generally $\endgroup$
    – glS
    Commented Jun 26, 2021 at 15:44

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