From: scerir (scerir@libero.it)
Date: Thu Sep 12 2002 - 08:37:56 MDT
Lee Corbin
> But to return to MWI again, probabilities there retain their
> classical simplicity.
In MWI? Difficult to normalize probabilities in there.
But MWI is very useful anyway. Because it goes deeper.
That is (perhaps, who knows) because QM is 'non-local'
and the MWI is 'non-local' by definition: *many* worlds!
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.
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.
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.
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 )
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*.
This archive was generated by hypermail 2.1.5 : Sat Nov 02 2002 - 09:16:59 MST