Re: Can we get any quantum state from some classical state by unitary

From: Tian-Ming Bu (tianming.bu_at_gmail.com)
Date: 01/13/05 

Date: Thu, 13 Jan 2005 19:29:00 +0000 (UTC)

I grately appreciate your reply! My research interesting is quantum
computing, but I major in computer science. So I am not familiar with
physics and I don't know whether my question is naive in physics.

Furthermore, I am not familiar with some notations and concepts in it.
Indeed, I can't well understand your reply. For example, what's the
meaning of [x,p] and <Delta x^2>? what's the meaning of x and p? In
quantum computing, there is no such notations. Can you explain them
more clearly?

Thanks.

Dan Platt wrote:
> That's a very neatly posed problem!
>
> BUT my first impression is that there is a problem representing the
> classical state, and the unitary operator that will take you between
> classical and quantum states.
>
> In the classical space, you would have [x',p'] = 0 (see below*); in
the
> quantum space, you have [x,p] = i h-bar. You want to find some
unitary
> operator U that will take you from the first space to the 2nd. Such
a
> unitary operator would take you from classical state |psi'> to |psi>
via
> (U^ = hermitian conjugate of U):
>
> |psi> = U|psi'>
>
> So, x = Ux'U^, p = Up'U^ and [x, p] = U[x', p']U^ = 0, which doesn't
work.
>
> Maybe another approach: Is it possible to construct a transformation
> between a WKB (sometimes considered "classical" -- but you might need
to
> determine in what sense) approximated state and the exact state it
> should represent? OR -- what conditions does Ehrenfest's theorem
hold
> well for, and what happens to them when they start to break down (in
> this case, you're trying to do a |"\psi" - "\phi"| on the classical
side
> rather than on the quantum side)?
>
> Tian-Ming Bu wrote:
> > Hi,
> >
> > Is it possible that given an quantum state \psi, you transform from
> > some classical state by unitary transformation to get another
quantum
> > state \phi, and |\psi - \phi| <= \epsilon? namely, the distance
between
> > the two quantum state is close.
> >
> > Thank you in advances.
> >
>
> * Generally,
>
> <Delta x^2><Delta p^2> >= |<[x, p] > |^2 / 4, so if a classical state
is
> one where <Delta x> = 0 and <Delta p> = 0, it follows that the two
> operators would have to commute in that representation.
>
> Dan

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