On 19/04/2016 12:17 am, Bruno Marchal wrote:
On 18 Apr 2016, at 09:45, Bruce Kellett wrote:
Let me reduce this to simple steps:
1) MWI is an interpretation of QM only. I.e., it reproduces all the
results of QM without adding any additional structure or dynamics.
What do you mean by QM? I am not sure I agree with you. Everett did
not talk about a new intepretation of QM, but about a new formulation
of QM. And he is right in the sense of the logician. Before Everett:
QM was formulated roughly SWE + Collapse + an implicit dualist theory
of mind or of scale (mircro/macro). Everett's QM is SWE, the abandon
of collapse, + a mechanist theory of mind, with the implicit use of
the FPI.
No, you are confusing the mathematical theory of QM with its various
interpretations. The mathematical theory of quantum mechanics is the
theory that associates physical states with vectors in Hilbert space;
observables with Hermitian operators in that space; and measurement
results with the eigenvalues of the corresponding operator. Supplemented
with the Born rule, which states that the probability for obtaining any
particular eigenvalue as the result of a measurement is given by the
absolute square of the coefficient of that eigenvalue in the
superposition describing the state, we get the standard mathematical
theory of quantum mechanics.
Note that this says nothing about collapse or not, about one or many
worlds, or about any interpretational issues. Interpretational issues
are overlaid on this basic theory of QM, and it is central to the whole
discussion that the predicted experimental results depend only on the
underlying theory and not on the interpretational superstructure. All
interpretations must give the same predictions for experimental results
or else they are alternative theories and not interpretations of
(standard) QM.
For a logician, if QM (without collapse) is formalized, you get an
"Herbrand minimal model" which contains already all relative state
(like we get them already in the sigma_1 arithmetic with the Mechanist
Hypothesis in the Cognitive Science).
Given the linearity of the tensor product and the evolution, we can
only abstract away the self-superposition, although we would have to
take them into account if we get a quantum brain (and here the SWE
give non ambigous result where a collapse theory has to first make
more precise how the (non local) collapse is made physically.
I don't know what you are talking about. But you are still confusing the
theory with its interpretation.
2) The QM state describing an entangled singlet pair does not refer
to, or depend on, the separation between the particles.
OK. But the singlet state describe an infinity of Bob and Alice with
their spin correlated, yet both of them see their own particles with a
random result, as none of them know in which universe they are. They
know only one thing for sure: their spin are correlated, and remains
so independently of the distance.
The only thing either of them knows for sure locally is that they have
50% probability of getting |+> and 50% probability of getting |->. After
the experiment is complete and completely decohered locally, they have
just one result. You might interpret this situation by claiming that
there are two local copies of each, one with |+> and one with |->, but
that is an interpretation, it is not the mathematical theory, which
predicts only the probabilities.
3) The quantum calculation of the joint probabilities depends on the
relative orientation between the separate measurements on the
separated particles.
No problem.
4) This quantum calculation is the same for any physical separation,
since the singlet state itself does not depend on the separation.
No problem.
5) The quantum calculation is, therefore, intrinsically non-local
because it does not depend on the separation, which can be
arbitrarily large.
This does not follow. It would be if the state |psi> = (|+>|-> -
|->|+>) would be interpreted by We know that Alice has a particle in
state |+> or in state |-> and Bob the opposite. But the state (|+>|->
- |->|+>) means eaxctly that neither Alice nor Bob know in which
universe they are. It could be one with |+'> or |-'> or whatever.
Again, you are confused by the basis issue. The singlet state can be
expanded in the above form in any basis whatsoever, sure, but you are
forgetting the crucial matter of contextuality. For Alice, sitting there
with her EPR particle, the basis in which the singlet state is expanded
is completely irrelevant -- there is only one Alice at this stage. Once
she chooses the setting for her magnet, she needs to be able to
calculate the probabilities for the possible results that she can get.
To calculate these probabilities she chooses the basis that corresponds
to the orientation of her magnet. This is the effect of contextuality.
She could, of course, choose to make her calculations in any of the
other infinite number of possible bases. But then, in order to relate
the results of her calculation to the results of her experiment, she has
to perform the rotation to bring the theoretical and experimental bases
into alignment.
6) Since MWI does not add anything to standard QM, and standard QM
gives a non-local account of the probabilities we are considering,
any MWI account must also be intrinsically non-local.
Proof?
The above sequence of steps constitutes the proof.
Don't invoke Bell's theorem, because it assumes Alice and Bob are in
the same reality, where without collapse, the measurement of Bob and
Alice propagate only locally from multiple Alice to mutiple Bob, as
describe by the superposition singlet state (in any base).
I don't have to invoke Bell's theorem. That is not relevant for what I
am saying above. Let me spell it out:
1) The QM singlet state |psi> = (|+>|-'> - |->|+'>) is intrinsically
non-local. This is universally agreed. If it were local, then QM would
be intrinsically local and we would not be having this discussion.
2) A hidden variable theory, which would be a "completion" of QM (in
Einstein's sense), would be a /different/ theory, not standard QM. (It
would have a mechanism whereby the orientation at A was available for
the interpretation of the orientation at B. This is the mechanism that
Jesse was confusing with the mathematical theory for calculating
probabilities. QM is non-local because it provides no such mechanism.
Any local theory must provide such a mechanism, and thus must be a
different theory from conventional QM.)
3) What Bell's argument shows is that even if you "complete" QM by
adding hidden variables, the resulting theory is still inescapably
non-local.
4) Experiment irrefutable bears out the violation of the Bell
inequalities, so demonstrates that reality is non-local.
If I find some time, I might try to describe this with the density
matrix formalism, which I think can make this more obvious.
One physical reality, and/or hidden variables specifying unqiueness of
state + Violation of Bell's inequality entails non-locality. That is
shown by Bell's inequality violatin.
But without "collapsing" a wave at a distance, the apparent
non-locality comes only from Alice or Bob determining in which
universe they are. There are just no reason they found themsleves in
the same universe. If they can compare the results, it is only after
the contagion of their superposed state with each other, and in that
case, the statistics implies the Bell correlations, without any
physical action at a distance. You need to transform the pure state in
some mixture, before the measurement to get non-locality, but such
mixture are local and different for each Alice and Bob in the
superposition state, so you cannot take them as definite like if Alice
or Bob could know that in advance.
What can I say? There are only two local possibilities for both Alice
and Bob: they get either |+> or |->, relative to their own context as
defined by the orientation of their respective magnets. So there can
only ever be four distinct worlds. You can multiply them as many times
as you like, as Jesse seems to want to do, but it is a basic fact of
statistics that if you take an ensemble of a very large number of a
limited set of states, any random selection from that ensemble will
result in one of the states that you originally put in -- nothing new is
generated. The four possible worlds arising from our scenario are
|+>|+'>, |+>|-'>, |->|+'>, and |->|-'>, no different combinations can
ever arise.
This crucial thing here, though, is that these four possibilities are
not equally probable. The distinct probabilities for each possible world
are given by the fundamental quantum calculation, which is independent
of the interpretation you overlay on the theory: the probabilities
depend only on the relative orientations of the magnets of Alice and Bob
-- they depend on nothing else. To get the *relative* orientation of the
magnets you have to compare the separate orientations for Alice and Bob
-- /at remote, potentially spacelike, separate locations/. That is why
QM is said to be non-local.
Nothing that you have said obviates anything in the above argument. The
conclusion is sound; you merely obfuscate by your talk of and indefinite
number of possible worlds. On further reflection, I think you are really
confused by the basis issue. I suggest that you think more carefully
about the role of contextuality in the selection of an appropriate
basis. You need only one basis for each experimenter -- all other bases
are irrelevant.
It is "shocking" because it is really the self-multiplication which
explains the apparent non-locality, but then that was also the case
for the apparent indeterminacy.
Put in a different way: when Alice and Bob make their measurement,
they might get result violating the correlation, but that would make
their belonging to different cross-product term of the final
superposition, so they would not been able to compare those forbidden
results.
There are no such cross terms in the expansion. They get only |+> or
|->, and any possible combination is allowed in general (unless the
theory gives an exactly zero probability for one or more of the four
possible worlds). Forbidden results are forbidden by the laws of
physics, and the indication that a particular combination is forbidden
is that the corresponding probability is calculated to be zero. So there
are no "forbidden cross-product terms". You are talking nonsense.
Bruce
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