On 27/04/2016 4:57 pm, Bruno Marchal wrote:
On 27 Apr 2016, at 06:49, Bruce Kellett wrote:
On 27/04/2016 1:51 pm, Brent Meeker wrote:
That's pretty much the many-universes model that Bruno proposes. But
it's non-local in the sense that the "matching scheme" must take
account of which measurements are compatible, i.e. it "knows" the
results even while they are spacelike separated.
Exactly, the model assumes the results it is trying to get. It is not
a local physical model because the statistics do not originate locally.
The statistic did originate locally. Alice and Bob did prepare the
singlet state locally, and then travel away.
That is not strictly correct. The singlet state is conventionally
prepared centrally between A and B so that the measurements can be made
at spacelike separation. That would not be possible if A and B jointly
prepare the state then move away.
They are in infinitely many worlds, and in each with opposite spin.
There are only two possible spin states for each -- so there are really
only two distinct possible worlds. Multiplying copies of these two does
not seem to accomplish much.
The cos^2(theta) is given by the math of the 1/sqrt(2)AB(I+>I-> -
I->I+>)) = 1/sqrt(2)ABI+>I-> - 1/sqrt(2)ABI->I+>. With your
explanation to Jesse, I keep the feeling that you talk like if Alice
or Bob reduce the wave after their measurement, but they just localize
themselves in the relative branches.
Certainly, the cos^2(theta/2) comes from applying the standard quantum
rules to the singlet state
|psi> = (|+>|-> - |->|+>)/sqrt(2) (adding AB to this state adds
nothing). I think it would be instructive to actually go through the
usual quantum derivation of the correlations because what you call
"reducing the wave after the measurement" is actually the result of
applying the standard quantum rules. It has nothing to do with so-called
'collapse' interpretations: it is simply in the theory.
Quantum rules for measurement say that the initial state can be expanded
in the basis corresponding to the particular measurement in question
(contextuality). That is what the state |psi> above is -- the quantum
expansion of the singlet state in the basis in which say Alice is doing
her measurement. Quantum rules then say that the result of the
measurement (after decoherence has fully operated) is one of the
eigenstates in the expansion, and the measurement result is the
corresponding eigenvalue. In our case, there are two possibilities for
Alice after her measurement is complete: result '+', with corresponding
eigenstate |+>|->, or '-', with corresponding eigenstate |->|+>. There
are no other possibilities, and Alice has a 50% chance of obtaining
either result, or of being in the corresponding branch of the evolved
wave function.
The question now arises as to how the formalism describes Bob's
measurement, assuming that it follows that of Alice (there will always
be a Lorentz frame in which that is true for spacelike separations. For
timelike separations, it is either true, or we reverse the A/B labels so
that it is true.) Since the description of the state does not depend on
the separation between A and B, after A gets '+' and her eigenstate is
|+>|->, Bob must measure the state |-> in the direction of his magnet.
To get the relative probabilities for his results, we must rotate the
eigenfunction from Alice's basis to the basis appropriate for Bob's
measurement. This is the standard rotation of a spinor, given by
|-> = sin(theta/2)|+'> -i cos(theta/2)|-'>
Applying the standard quantum rules to this state, Bob has a probability
of sin^2(theta/2) of obtaining a '+' result, and a probability of
cos^2(theta/2) of obtaining a '-' result.
Using test values for the relative orientation, theta, we get the usual
results. For theta = 0º, Bob has probability 0 of obtaining '+', and
probability 1 of obtaining '-'. For 90º orientation, the probabilities
for '+' and '-' are both 0.5. For a relative orientation of 120º, Bob's
probability of getting '+' is 0.75 and the probability of getting '-' is
0.25. And so on for the familiar results.
This is not controversial, and the result depends only on the standard
rules of quantum mechanics. The problem of interpretation, of course, is
that since Alice and Bob are at different locations, and the state they
are measuring is independent of separation, there is an intrinsic
non-locality implied by the standard calculation. If you take out the
quantum rule that the result of a measurement is, after decoherence,
reduction to an eigenstate with the corresponding eigenvalue, you take
away an essential ingredient of the quantum derivation, and leave Bob's
measurement as being completely independent of that of Alice, so the
only possible results for Bob are '+' and '-' with equal probability,
whatever the orientation of his magnet.
Any account that deviates from this is no longer a standard quantum
account because it would not conform to the above rules. And these rules
are among the best-tested rules in all of physics. They are the basis
for the whole of the phenomenal success of this theory over nearly a
hundred years and in every field in which it has been applied. You
abandon these principles only at extreme peril.
Like Jesse said: no "matching" between copies of measurement-outcomes
at different locations takes place at any location in space-time that
doesn't lie in the future light cone of both measurements. Only if a
reduction of the wave occur would a genuine action at a distance have
to take place to keep up the cos^2(theta). In the MWI, we keep it
intact because 1/sqrt(2)ABI+>I-> - 1/sqrt(2)ABI->I+> describes a
global state of the multiverse. There is a form on non separability,
but it does not use non local action. It uses only the fact that the
many Alice and Bob are in the same branches and remains in the same
branches when travelling away of each other in each branch, but they
both cannot know in which branch they are, and what is the spin of
their respective particles. They do know that they are correlated by
1/sqrt(2)ABI+>I-> - 1/sqrt(2)ABI->I+>, but that is all they can know.
Frankly, I do not know what this means. I think that you will have to
work through the details more explicitly. You have to show where the
standard rules of quantum mechanics cease to apply, and why. And why
they cease only for this entangled state, while remaining intact
elsewhere. There seem to be questions of consilience and consistency at
stake here.
Bruce
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