From: Lee Corbin (lcorbin@tsoft.com)
Date: Thu Sep 19 2002 - 04:16:52 MDT
Serafino (scerir) wrote on September 12
> The superposition principle requires non-local states.
> The nature of this original non-locality is kinematical,
> not dynamical, as D. Zeh pointed out. But we use to think
> in terms of dynamical non-local effects, such as spooky
> actions at-a-distance, and so on.
(Serafino---sorry that I'm about a week behind answering
your posts, and about 3 of *your* posts behind, but I will
catch up.)
> Actually this kinematical non-local reality (i.e. entanglements)
> exists *before* any related local events (spatially separated)
> occur. In quantum teleportation a state has to be prepared
> initially. But after this preparation the state exists but it is at no
> place. Look.
>
> Alice has photon 1, which is in a certain quantum state,
> unknown to Alice and unknown to anyone else.
> Let us say that this unknown quantum state is
> |psi>_1 = a |0>_1 + b |1>_1
> with |a|^2 + |b|^2 = 1
> and where |0>_1 and b |1>_1 represent two orthogonal quantum
> states and a and b represent complex amplitudes.
As I wrote before, *this* so far is clear and very standard.
But
> Now Alice wants to *transfer* her quantum state to Bob,
> which is remote, so she can not directly deliver it to him. But, fortunately,
> Alice also has a pair of entangled photons, let us say the photon 2 and
> the photon 3, and she already gave the photon 3 to Bob, who still has
> this particle.
Yes. I recall studying quantum teleportation a few years ago,
and thinking that I understood it, but your notation following
really floors me:
> Leaving apart normalization factors we can write that the total state
> of those 3 photons is
>
> |psi>_1,2,3 =
>
> ( |0>_1 |1>_2 - |1>_1 |0>_2 ) (- a |0>_3 - b |1>_3 ) +
> ( |0>_1 |1>_2 + |1>_1 |0>_2 ) (- a |0>_3 + b |1>_3 ) +
> ( |0>_1 |0>_2 - |1>_1 |1>_2 ) ( a |1>_3 + b |0>_3 ) +
> ( |0>_1 |0>_2 + |1>_1 |1>_2 ) ( a |1>_3 - b |0>_3 )
An excellent paper has helped me to understand what you are saying here.
http://www.physics.ohio-state.edu/~wilkins/writing/Assign/topics/Q-trans-prl.pdf
Of the four terms comprising eight factors, the second factor above
(- a |0>_3 - b |1>_3 )
represents the contribution from Bob's particle (the one of the
two entangled photons, photon 3). But the first factor
( |0>_1 |1>_2 - |1>_1 |0>_2 )
is what? It *looks* like one component of the superposition of particles
1 and 2 (with the other three components lying directly underneath in the
other terms). Are you here using the four orthonormal Bell states for the
particles 1 and 2? The development in the above quoted paper seemed
a little easier to follow, but I would still like to understand your
equation here. Am I on the right track?
(More later. I have had extremely little time to deal with email
this week and last.)
> Alice now performs a measurement on photons 1 and 2 and she projects her
> two photons onto *one* of these four states below:
> ( |0>_1 |1>_2 - |1>_1 |0>_2 )
> ( |0>_1 |1>_2 + |1>_1 |0>_2 )
> ( |0>_1 |0>_2 - |1>_1 |1>_2 )
> ( |0>_1 |0>_2 + |1>_1 |1>_2 )
>
> And consequently Bob will found his photon in *one* of these four states below
> (- a |0>_3 - b |1>_3 )
> (- a |0>_3 + b |1>_3 )
> ( a |1>_3 + b |0>_3 )
> ( a |1>_3 - b |0>_3 )
>
> Now Alice, who wants to *transfer* the unknown quantum state of photon
> 1 to Bob, must inform Bob, via a classical channel, about her measurement
> (projection) result (on photons 1 and 2). So Bob can perform (25% of times
> it is not required) the right simple unitary transformation on his photon 3, in
> order to obtain the initial quantum state |psi>_1 = a |0>_1 + b |1>_1
>
> Note that Alice does not get any information, from her measurement, about
> the quantum state she wants to *transfer* and about the values of those
> a and b amplitudes. Note also that during Alice's measurement photon
> 1 loses his original quantum state, as required by the no-cloning theorem.
>
> Ok, that was the basic teleportation of a quantum state from Alice to Bob.
>
> Now something *very* strange happens in the MWI version. Alice's
> measurement does *not* project the superposition of
> ( |0>_1 |1>_2 - |1>_1 |0>_2 )
> ( |0>_1 |1>_2 + |1>_1 |0>_2 )
> ( |0>_1 |0>_2 - |1>_1 |1>_2 )
> ( |0>_1 |0>_2 + |1>_1 |1>_2 )
> onto just *one* of these quantum states (above). They *all* exist.
>
> And *all* these (Bob's) quantum states (below) also exist.
> (- a |0>_3 - b |1>_3 )
> (- a |0>_3 + b |1>_3 )
> ( a |1>_3 + b |0>_3 )
> ( a |1>_3 - b |0>_3 )
> and one of them (1 over 4 = 25% of times) is the same quantum state
> that Alice wanted to *transfer* to Bob. But it was *already* *there*.
I bet the upshot will be that the MWI version which *does* appear
to have simpler equations---but then I am an Everettista in your
cute phrase ;-) --- will be just as satisfactory. But I still
want to understand more of this amazing means of comparing the
two interpretations.
Lee
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