From: Lee Corbin (lcorbin@tsoft.com)
Date: Thu Sep 19 2002 - 22:36:23 MDT
Serafino writes
> In standard QM an instantaneous transfer of 'information' is not
> possible. Philippe Eberhard ('Bell's theorem and the Different
> Concepts of Locality', Nuovo Cimento, 46B, (1978), 392-419)
> showed that. Actually he built a theorem, using a rather
> mathematical definition of locality.
Well, I would say that in *any* QM an instantaneous transfer
of information is impossible. But maybe there are bizarre
theories...
> But in standard QM it *is* possible to 'teleport' some
> 'information' (actually a quantum state, unknown too)
> by exploiting EPR correlations, i.e. two entangled particles.
I think that you should use "influence", not information.
Otherwise, it directly contradicts your very first sentence
above.
> Notice that a
> quantum teleporting machine is the sum of two different
> impossible machines: a classical teleporting machine, and
> a quantum John Bell's telephone (a device using EPR pairs
> to send FTL messages). It is weird, imo, that two impossible
> machines, when combined, give a possible machine.
That's a very interesting observation. And it probably also
helps the student of all this recall how it works.
> Now Alice wishes to give Bob the information about a quantum system,
> i.e. the particle 1 prepared in an unknown quantum state. Alice lets this
> particle interact with a couple of entangled particles, 2 and 3, of which
> one, particle 3, was previously given to Bob and the other, particle
> 2, is used by her.
>
> The unknown state of particle 1 has the form
> |1> = a |+,1> + b |-,1>
Yes, this is very clear, of course. Now I'm writing for
others who haven't studied any quantum mechanics at all.
State 1, as Serafino has written it, using the "ket"
notation |1> is a complex number called an amplitude
and it's sort of an axiom of quantum mechanics (I think)
that it is also "superposition" of two other possible
states and can be written as such.
> EPR state of particles 2 and 3 has the form
> |2&3> = |+,2>|-,3> - |-,2>|+,3>
>
> The complete state of particles 1, 2 and 3 has the form
> |1&2&3> = a |+,1>|+,2>|-,3> - a |+,1>|-,2>|+,3> +
> b |-,1>|+,2>|-,3> - b |-,1>|-,2>|+,3>
Yes. Thank you. This is clearer than before; and
for those knowing even less than I, you can get this
equation by using the distributive law (just multiply
|1> above by |2&3> above. It's easy, if you don't
let the strangeness of the notation get to you.
> which can also be written as
> |1,2, one> * (-a |+,3> - b |-,3>) +
> |1,2, two> * (-a |+,3> + b |-,3>) +
> |1,2,three> * (a |-,3> + b |+,3>) +
> |1,2, four> * (a |-,3> - b |+,3>)
Here is where you lose me, again. (I realize that I should
go over to a table, sit down, right all this down and think
about it for half an hour. But I don't feel like it. I've
already stared at this for ten minutes, and---besides changing
a typo of yours where you had a "." instead of a ","
---am not getting anywhere.)
What are "one", "two", "three", and "four"? I think that
you have factored out the |+,3> and |-,3> some way, but
I don't see how. Is it possible to explain how say the
first term, |1,2,one> * (-a |+,3> - b |-,3>) arises?
If not :-( then I'll just have to go sit down for half
an hour, if that's the only way to really get it. (Thanks,
either way.)
Perhaps I can follow the rest of your explanation if
you can get me (or us, perhaps) past that.
Lee
> Now in MWI Alice performs a measurement on particles 1 and 2
> (at her side) and she obtains *all* these four possible states
> |1,2, one>, |1,2, two>, |1,2,three>, |1,2, four>
>
> This means that Bob, at his side, obtains, for the quantum state
> of particle 3, which is the hands of Bob, all these four possible
> states
> (-a |+,3> - b |-,3>)
> (-a |+,3> + b |-,3>)
> (a |-,3> + b |+,3>)
> (a |-,3> - b |+,3>)
>
> Actually, in MWI, the measurement performed by Alice creates
> four different worlds. In each of these worlds we can
> find just one Alice and just one of these Alice's outcomes
> |1,2, one>, |1,2, two>, |1,2,three>, |1,2, four>
> and we can also find just one Bob and just one of these
> states below, of particle 3, each one linked to an Alice's
> outcome
> (-a |+,3> - b |-,3>)
> (-a |+,3> + b |-,3>)
> (a |-,3> + b |+,3>)
> (a |-,3> - b |+,3>)
>
> Now notice that *one* of these four states is the *same* state
> of the particle 1, except for an irrelevant phase factor.
>
> Hence we can say that in MWI there is one world
> (over four) in which Alice teleports, by quantum means,
> instantaneously, without any classical channel, an unknown
> quantum state, to Bob.
>
> To say that Alice teleports instantaneously a quantum state
> (which is information) might also mean that the quantum
> state was already there, at Bob's side, in that particular
> world.
>
> Of course it is not impossible to build a model of an
> asymmetrical splitting of worlds - a world has a 'large'
> measure, another has a 'smaller' measure. It is also
> possible to use the entanglement-swapping procedure,
> which is much more powerful, hence build a net of .....
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