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
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
So MWI does not give a local account of the EPR results on entanged states.
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