From: Lee Corbin (lcorbin@tsoft.com)
Date: Sat Sep 21 2002 - 14:33:17 MDT
Hal writes
> Regarding Serafino's explanation of quantum teleportation:
>
> > 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>
...
> I think the idea is that "one", "two", "three" and "four" represent the
> four possible outcomes of a joint measurement on particles 1 and 2. Based
> on some other references, these are of the form (ignoring constants):
>
> |1,2, one> = |+,1>|-,2> - |-,1>|+,2>
> |1,2, two> = |+,1>|-,2> + |-,1>|+,2>
> |1,2, three> = |+,1>|+,2> - |-,1>|-,2>
> |1,2, four> = |+,1>|+,2> + |-,1>|-,2>
> [Serafino had written]
> > 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>)
> I think that works out algebraically.
Okay, let's see: if we take just the first product
you refer to, namely,
|1,2, one> * (-a |+,3> - b |-,3>)
we get (|+,1>|-,2> - |-,1>|+,2>) times (-a |+,3> - b |-,3>)
which by the distributive law for (X-Y)*(A-B) gives
-a |+,1>|-,2>|+,3> plus -b |+,1>|-,2>|-,3>
plus +a |-,1>|+,2>|+,3> plus +b |-,1>|+,2>|-,3>
Then, when you do it for the other three terms too,
you get 16 terms altogether, and exactly half of them
cancel out. (For instance, the 2nd and 3rd in my last
expression there end up going away.) Whew!
Then! Since only 8 are left, you still have to spot
that each one has its exact double, and, since we
are ignoring normalization factors we may write 2=1 ;-)
and so there are finally only 4 left. Double whew!
And those four are, by God,
> > |1&2&3> = a |+,1>|+,2>|-,3> - a |+,1>|-,2>|+,3> +
> > b |-,1>|+,2>|-,3> - b |-,1>|-,2>|+,3>
exactly as Serafino had said. (I finally had to resort
to paper and pencil. I *thought* that I was good at
mental algebra!!) This is the sort of thing that looks
as though if it's right at all, then it's simple. But no.
> In the MWI sense this measurement splits the world along the lines above,
> so that Bob's remote photon 3 ends up in one of those four states in each
> of the four different newly-separated "worlds". Serafino points out that
> in one of the worlds this means that Bob ends up with a copy of photon 1
> seemingly at the instant the measurement is made.
Okay, Hal, I'm starting to get it, and I sort of understand
your last paragraph. But now that the algebra is clear, I'll
go back to Serafino's posts and (also using your paragraph
here) perhaps get to a clear understanding of the whole thing.
Thanks,
Lee
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