The mistake made is to invoke classical reasoning after the measurements
are made. If the choice for the orientation of the polarizers were not
made in advance, then Alice and Bob cannot have said to have made any
definite choices at all. In some particular sector where Alice made some
particular result and found some particular result, she knows that Bob's
spin state. But Bob lives in larger sector of the multiverse which
includes sectors where Alic had made different choices.
Alice and Bob communicating later is not some trivial exchange of
information that existed a priori, it leads to a further de-facto
collapse of the wavefunction.
There isn't anything more to this that Alice measuring the spin of an
electron in a lab, and then letting Bob who doesn't know what direction
the spin was measured in, doing another measurement.
Saibal
On 16-04-2016 02:33, Bruce Kellett wrote:
On 16/04/2016 12:20 am, Bruno Marchal wrote:
On 14 Apr 2016, at 14:31, Bruce Kellett wrote:
Although all possible combinations of measurement outcomes exist
in MWI, it is not clear what limits the results of the two
observers to agree with quantum mechanics when they meet up in
just one of the possible worlds.
Because they have separated locally, and Alice's measurement just
inform both of them (directly for Alice and indirectly for Bob once
some classical bit of information is communicated by Alice to Bob by
the usual means).
This is the purported solution given by Deutsch and Hayden, amongst
many others. Unfortunately, it does not work, as can be demonstrated
by working through a specific example.
Consider the usual case of a spin singlet that splits into two
spin-half components that separate and are measured by A and B at
spacelike separation. There are two possible measurement results for
each observer, call them |+> and |->. The entangled state can then be
written as:
|psi> = (|+>|-> - |->|+>).
ignoring normalization factors for simplicity. The first ket applies
to observer A and the second to observer B.
This is the general expression for the singlet state in any basis,
such as would be define by the orientation of the measuring magnets.
We denote the measurement results in some other direction as |+'> and
|-'>.
A and B perform their measurements at spacelike separation, but each
chooses the measurement orientation outside the light cone of the
other. There are four possible combinations of results, corresponding
to four worlds in the MWI: |+>|+'>, |+>|-'>, |->|+'>, and |->|-'>.
Since each observer has a 50% chance of getting |+> and 50% of getting
|->, and the two measurements are completely independent of each
other, it would seem that each of these four worlds is equally likely.
But this conclusion is contradicted by quantum mechanics: if the two
observers, by chance, have their magnets aligned, then the |+>|+'> and
|->|-'> combinations are impossible. In general, the probabilities of
the four possible joint outcomes depend explicitly on the relative
orientation of the magnets of the A and B -- they are seldom all
equal. How is this taken into account in the formalism?
In the formalism of QM, the answer is clear enough. Given the
expression for |psi> in an arbitrary basis, as above, we can choose
the basis for this expansion to be that for the orientation of magnet
A. But then, in order to get the relevant outcomes for B, we have to
rotate this expansion to the basis corresponding to the orientation of
magnet B. But we have to do this rotation before B makes his
measurement! How does B know the necessary rotation angle? Recall that
both A and B make independent arbitrary rotations at spacelike
separations.
After the measurements are complete, A and B communicate their
results to each other, so the branch of B that measured |+'>
communicates this to both copies of A, to get the combinations |+>|+'>
and |->|+'>. Similarly, the branch of B that got the result |-'>
communicates this to both copies of A, to get the remaining two
combinations |+>|-'> and |->|-'>. Deutsch and Hayden propose that
non-locality is eliminated by B communicating his orientation angle as
well as his result to A. But adding the angle theta to the information
transmitted does not change the fact that one copy of B transmits a
|+'> result and one copy transmits a |-'> result. In other words, this
extra orientation information is completely irrelevant to the outcomes
of the measurements, and also irrelevant to the relatives
probabilities for the our possible worlds.
Deutsch and Hayden have not shown that this EPR experiment is local
in MWI -- they still have to use the rotation of the wave function
basis for B's measurement _before_ that measurement is made, and that
information is not locally available to B, it can only have been
transmitted non-locally.
So MWI does not give a local account of the EPR results on entanged
states.
Bruce
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it,
send an email to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to
everything-list@googlegroups.com.
Visit this group at https://groups.google.com/group/everything-list
[1].
For more options, visit https://groups.google.com/d/optout [2].
Links:
------
[1] https://groups.google.com/group/everything-list
[2] https://groups.google.com/d/optout
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
