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

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