Re: Quantum states from classical states?



> > If
> > anybody wants to learn more about how to distinct those states, he may
> > want to take a look at Phys. Rev. Lett. 94, 153601 (2005)
>
> This sounds interesting. If you can give an equivalent Arxiv preprint
> reference that would be appreciated.

Sure, it´s quant-ph/0408029

> In the meantime, allow me to point out that with a different ad-hoc
> assumption -- namely, that one uses symmetrically ordered operators when
> calculating averages -- one finds that the quantum states that "behave
> clasically" are precisely those with positive Wigner distributions.

At a first glance it seemed dissapointing to me that the property of
"being classical" depends on the ordering (we have chosen normal as the
most commonly used), but at the end why there should be just one way of
embedding classical statistics into quantum? Why there should be such
an embedding at all? the most important thing is to receive classical
statistics in the limit \hbar -> 0. I´m prepearing a short paper on
that.

> > To be more precise, only hamiltonians at most quadratic in x, p leave
> > coherent states invariant.

> And -- for pure states -- it is precisely these Hamiltonians that
> generate unitary transformations that leave the property of having
> a positive Wigner distribution unaltered. These transformations are
> just linear canonical transformations.

Yes, the quadratic hamiltonians generate the representation of the
metaplectic group - a universal double cover of the symplectic group.

> The pure states with positive
> Wigner distributions include coherent states, squeezed coherent states,
> and (if one allows the term "state" to be abused somewhat) states with
> a definite value of Ux + Vp where U and V are real c-numbers.

Let me just, very gently (dont want to go into wars :-), remark that
squeezing may be rather regarded as a manifestation of a quantum
property. A more or less systematic way of treating squeezing as a
signature of entanglement for finite systems may be found here:
quant-ph/0504005 and Refs. therein.

Best,
jarek


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