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Summary: Describes the measurement problem in quantum mechanics          and physical and metaphysical approaches to its solution.
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Posted-By: auto-faq 3.3 (Perl 5.003)
Archive-name: physics-faq/measurement-in-qm

  Measurement in quantum mechanics FAQ
  Maintained by Paul Budnik, [e-mail], http://www.mtnmath.com

  This FAQ describes the measurement problem in QM and approaches to its
  solution. Please help make it more complete. See ``What is needed''
  for details.    Web version: http://www.mtnmath.com/faq/meas-qm.html

  1.  About this FAQ

  The general sci.physics FAQ does a good job of dealing with technical
  questions in most areas of physics. However it has no material on
  interpretations of QM which are among the most frequently discussed
  topics in sci.physics. Hence there is a need for this supplemental
  FAQ.

  This document is probably out of date if you are reading it more than
  30 days after the date which appears in the header.

  This FAQ is on the web at: http://www.mtnmath.com/faq/meas-qm.html

  You can get it by e-mail or FTP from rtfm.mit.edu.

  By FTP, look for the file:

  /pub/usenet/news.answers/physics-faq/measurement-in-qm

  By e-mail send a message to [e-mail] with a blank
  subject line and the words:

  send usenet/news.answers/physics-faq/measurement-in-qm

  The main sci.physics FAQ is in this same directory with file names
  part1 through part4 and can be retrieved in the same way.  You can put
  multiple send lines in a single e-mail request.

  This document, as a collection, is Copyright 1995 by Paul P. Budnik
  ([e-mail]).  The individual articles are Copyright 1995 by the
  individual authors listed.  All rights are reserved.    Permission to
  use, copy and distribute this unmodified document by any means and for
  any purpose EXCEPT PROFIT PURPOSES is hereby granted, provided that
  both the above Copyright notice and this permission notice appear in
  all copies of the FAQ itself.    Reproducing this FAQ by any means,
  included, but not limited to, printing, copying existing prints,
  publishing by electronic or other means, implies full agreement to the
  above non-profit-use clause, unless upon explicit prior written
  permission of the authors.

  This FAQ is provided by the authors ``as is''. with all its faults.
  Any express or implied warranties, including, but not limited to, any
  implied warranties of merchantability, accuracy, or fitness for any
  particular purpose, are disclaimed.  If you use the information in
  this document, in any way, you do so at your own risk.

  2.  The measurement problem

  Paul Budnik [e-mail]

  The formulation of QM describes the deterministic unitary evolution of
  a wave function. This wave function is never observed experimentally.
  The wave function allows us to compute the probability that certain
  macroscopic events will be observed. There are no events and no
  mechanism for creating events in the mathematical model. It is this
  dichotomy between the wave function model and observed macroscopic
  events that is the source of the interpretation issue in QM. In
  classical physics the mathematical model talks about the things we
  observe.  In QM the mathematical model by itself never produces
  observations.    We must interpret the wave function in order to relate
  it to experimental observations.

  It is important to understand that this is not simply a philosophical
  question or a rhetorical debate. In QM one often must model systems as
  the superposition of two or more possible outcomes. Superpositions can
  produce interference effects and thus are experimentally
  distinguishable from mixed states. How does a superposition of
  different possibilities resolve itself into some particular
  observation? This question (also known as the measurement problem)
  affects how we analyze some experiments such as tests of Bell's
  inequality and may raise the question of interpretations from a
  philosophical debate to an experimentally testable question. So far
  there is no evidence that it makes any difference. The wave function
  evolves in such a way that there are no observable effects from
  macroscopic superpositions. It is only superposition of different
  possibilities at the microscopic level that leads to experimentally
  detectable interference effects.

  Thus it would seem that there is no criterion for objective events and
  perhaps no need for such a criterion. However there is at least one
  small fly in the ointment. In analyzing a test of Bell's inequality
  one must make some determination as to when an observation was
  complete, i. e. could not be reversed. These experiments depend on the
  timing of macroscopic events. The natural assumption is to use
  classical thermodynamics to compute the probability that a macroscopic
  event can be reversed. This however implies that there is some
  objective process that produces the particular observation. Since no
  such objective process exists in current models this suggests that QM
  is an incomplete theory.  This might be thought of as the Einstein
  interpretation of QM, i. e., that there are objective physical
  processes that create observations and we do not yet understand these
  processes.  This is the view of the compiler of this document.

  For more information:

  Ed. J. Wheeler, W. Zurek, Quantum theory and measurement, Princeton
  University Press, 1983.

  J. S. Bell, Speakable and unspeakable in quantum mechanics, Cambridge
  University Press, 1987.

  R.I.G. Hughes, The Structure and Interpretation of Quantum Mechanics,
  Harvard University Press, 1989.

  3.  Schrodinger's cat

  Paul Budnik [e-mail]

  In 1935 Schrodinger published an essay describing the conceptual
  problems in QM1. A brief paragraph in this essay described the cat
  paradox.
     One can even set up quite ridiculous cases. A cat is penned up
     in a steel chamber, along with the following diabolical device
     (which must be secured against direct interference by the cat):
     in a Geiger counter there is a tiny bit of radioactive
     substance, so small that perhaps in the course of one hour one
     of the atoms decays, but also, with equal probability, perhaps
     none; if it happens, the counter tube discharges and through a
     relay releases a hammer which shatters a small flask of
     hydrocyanic acid. If one has left this entire system to itself
     for an hour, one would say that the cat still lives if meanwhile
     no atom has decayed.  The first atomic decay would have poisoned
     it. The Psi function for the entire system would express this by
     having in it the living and the dead cat (pardon the expression)
     mixed or smeared out in equal parts.

     It is typical of these cases that an indeterminacy originally
     restricted to the atomic domain becomes transformed into
     macroscopic indeterminacy, which can then be resolved by direct
     observation. That prevents us from so naively accepting as valid
     a ``blurred model'' for representing reality. In itself it would
     not embody anything unclear or contradictory. There is a
     difference between a shaky or out-of-focus photograph and a
     snapshot of clouds and fog banks.

  We know that superposition of possible outcomes must exist
  simultaneously at a microscopic level because we can observe
  interference effects from these.  We know (at least most of us know)
  that the cat in the box is dead, alive or dying and not in a smeared
  out state between the alternatives. When and how does the model of
  many microscopic possibilities resolve itself into a particular
  macroscopic state? When and how does the fog bank of microscopic
  possibilities transform itself to the blurred picture we have of a
  definite macroscopic state.  That is the measurement problem and
  Schrodinger's cat is a simple and elegant explanations of that
  problem.

  References:

  1 E. Schrodinger, ``Die gegenwartige Situation in der
  Quantenmechanik,'' Naturwissenschaftern. 23 : pp. 807-812; 823-823,
  844-849. (1935).  English translation: John D. Trimmer, Proceedings of
  the American Philosophical Society, 124, 323-38 (1980), Reprinted in
  Quantum Theory and Measurement, p 152 (1983).

  4.  The Copenhagen interpretation

  Paul Budnik [e-mail]

  This is the oldest of the interpretations. It is based on Bohr's
  notion of `complementarity'. Bohr felt that the classical and quantum
  mechanical models were two complementary ways of dealing with physics
  both of which were necessary. Bohr felt that an experimental
  observation collapsed or ruptured (his term) the wave function to make
  its future evolution consistent with what we observe experimentally.
  Bohr understood that there was no precise way to define the exact
  point at which collapse occurred. Any attempt to do so would yield a
  different theory rather than an interpretation of the existing theory.
  Nonetheless he felt it was connected to conscious observation as this
  was the ultimate criterion by which we know a specific observation has
  occurred.

  References:

  N. Bohr, The quantum postulate and recent the recent development of
  atomic theory, Nature, 121, 580-89 (1928), Reprinted in Quantum Theory
  and Measurement, p 87, (1983).

  5.  Is QM a complete theory?

  Paul Budnik [e-mail]

  Einstein did not believe that God plays dice and thought a more
  complete theory would predict the actual outcome of experiments.  He
  argued1 that quantities that are conserved absolutely (such as
  momentum or energy) must correspond to some objective element of
  physical reality. Because QM does not model this he felt it must be
  incomplete.

  It is possible that events are the result of objective physical
  processes that we do not yet understand. These processes may determine
  the actual outcome of experiments and not just their probabilities.
  Certainly that is the natural assumption to make. Any one who does not
  understand QM and many who have only a superficial understanding
  naturally think that observations come about from some objective
  physical process even if they think we can only predict probabilities.

  There have been numerous attempts to develop such alternatives.  These
  are often referred to as `hidden variables' theories. Bell proved that
  such theories cannot deal with quantum entanglement without
  introducing explicitly nonlocal mechanisms2.    Quantum entanglement
  refers to the way observations of two particles are correlated after
  the particles interact. It comes about because the conservation laws
  are exact but most observations are probabilistic.  Nonlocal
  operations in hidden variables theories might not seem such a drawback
  since QM itself must use explicit nonlocal mechanism to deal with
  entanglement. However in QM the non-locality is in a wave function
  which most do not consider to be a physical entity. This makes the
  non-locality less offensive or at least easier to rationalize away.

  It might seem that the tables have been turned on Einstein. The very
  argument he used in EPR to show QM must be incomplete requires that
  hidden variables models have explicit nonlocal operations. However it
  is experiments and not theoretical arguments that now must decide the
  issue. Although all experiments to date have produced results
  consistent with the predictions of QM, there is general agreement that
  the existing experiments are inconclusive3. There is no conclusive
  experimental confirmation of the nonlocal predictions of QM. If these
  experiments eventually confirm locality and not QM Einstein will be
  largely vindicated for exactly the reasons he gave in EPR. Final
  vindication will depend on the development of a more complete theory.

  Most physicists (including Bell before his untimely death) believe QM
  is correct in predicting locality is violated. Why do they have so
  much more faith in the strange formalism of QM than in basic
  principles like locality or the notion that observations are produced
  by objective processes? I think the reason may be that they are
  viewing these problems in the wrong conceptual framework. The term
  `hidden variables' suggests a theory of classical-like particles with
  additional hidden variables. However quantum entanglement and the
  behavior of multi-particle systems strongly suggests that whatever
  underlies quantum effects it is nothing like classical particles.  If
  that is so then any attempt to develop a more complete theory in this
  framework can only lead to frustration and failure.  The fault may not
  be in classical principles like locality or determinism. They failure
  may only be in the imagination of those who are convinced that no more
  complete theory is possible.

  One alternative to classical particles is to think of observations as
  focal points in state space of nonlinear transformations of the wave
  function. Attractors in Chaos theory provide one model of processes
  like this. Perhaps there is an objective physical wave function and QM
  only models the average or statistical behavior of this wave function.
  Perhaps the structure of this physical wave function determines the
  probability that the wave function will transform nonlinearly at a
  particular location. If this is so then probability in QM combines two
  very different kinds of probabilities. The first is the probability
  associated with our state of ignorance about the detailed behavior of
  the physical wave function. The second is the probability that the
  physical wave function will transform with a particular focal point.

  A model of this type might be able to explain existing experimental
  results and still never violate locality. I have advocated a class of
  models of this type based on using a discretized finite difference
  equation rather then a continuous differential equation to model the
  wave function4. The nonlinearity that must be introduced to discretize
  the difference equation is a source of chaotic like behavior.    In this
  model the enforcement of the conservation laws comes about through a
  process of converging to a stable state. Information that enforces
  these laws is stored holographic-like over a wide region.

  Most would agree that the best solution to the measurement problem
  would be a more complete theory. Where people part company is in their
  belief in whether such a thing is possible. All attempts to prove it
  impossible (starting with von Neumann5) have been shown to be flawed6.
  It is in part Bell's analysis of these proofs that led to his proof
  about locality in QM. Bell has transformed a significant part of this
  issue to one experimenters can address. If nature violates locality in
  the way QM predicts then a local deterministic theory of the kind
  Einstein was searching for is not possible. If QM is incorrect in
  making these predictions then a more accurate and more complete theory
  is a necessity. Such a theory is quite likely to account for events by
  an objective physical process.

  References: 1 A. Einstein, B. Podolsky and N. Rosen, Can quantum-
  mechanical descriptions of physical reality be considered complete?,
  Physical Review, 47, 777 (1935).  Reprinted in Quantum Theory and
  Measurement, p. 139, (1987).

  2 J. S. Bell, On the Einstein Podolosky Rosen Paradox, Physics, 1,
  195-200 (1964).  Reprinted in Quantum Theory and Measurement, p. 403,
  (1987).

  3 P. G. Kwiat, P. H. Eberhard, A. M. Steinberg, and R. Y. Chiao,
  Proposal for a loophole-free Bell inequality experiment, Physical
  Reviews A,  49, 3209 (1994).

  4 P. Budnik, Developing a local deterministic theory to account for
  quantum mechanical effects, hep-th/9410153, (1995).

  5 J. von Neumann, The Mathematical Foundations of Quantum Mechanics,
  Princeton University Press, N. J., (1955).

  6 J. S. Bell, On the the problem of hidden variables in quantum
  mechanics, Reviews of Modern Physics, 38, 447-452, (1966).  Reprinted
  in Quantum Theory and Measurement, p. 397, (1987).

  6.  The shut up and calculate interpretation

  Paul Budnik [e-mail]

  This is the most popular of interpretations. It recognizes that the
  important content of QM is the mathematical models and the ability to
  apply those models to real experiments. As long as we understand the
  models and their application we do not need an interpretation.

  Advocates of this position like to argue that the existing framework
  allows us to solve all real problems and that is all that is
  important.  Franson's analysis  of Aspect's experiment1 shows this is
  not entirely true.  Because there is no objective criterion in QM for
  determining when a measurement is complete (and hence irreversible)
  there is no objective criterion for measuring the delays in a test of
  Bell's inequality.  If the demise of Schrodinger's cat may not be
  determined until someone looks in the box (see item 2) how are we to
  know when a measurement in tests of Bells inequality is irreversible
  and thus measure the critical timing in these experiments?

  References:

  1 J. D. Franson, Bell's Theorem and delayed determinism, Physical
  Review D, 31,    2529-2532, (1985).

  7.  Bohm's theory

  Paul Budnik [e-mail]

  Bohm's interpretation is an explicitly nonlocal mechanistic model.
  Just as Bohr saw the philosophical principle of complementarity as
  having broader implications than quantum mechanics Bohm saw a deep
  relationship between locality violation and the wholeness or unity of
  all that exists. Bohm was perhaps the first to truly understand the
  nonlocal nature of quantum mechanics. Bell acknowledged the importance
  of Bohm's work in helping develop Bell's ideas about locality in QM.

  References: D. Bohm, A suggested interpretation of quantum theory in
  terms of "hidden" variables I and II, Physical Review,85, 155-93
  (1952).  Reprinted in Quantum Theory and Measurement, p. 369, (1987).

  D. Bohm & B.J. Hiley, The Undivided Universe: an ontological
  interpretation of quantum theory (Routledge: London & New York, 1993).

  Recently there has been renewed interest in Bohmian mechanics.  D.
  D"urr, S. Goldstein, N Zanghi, Phys. Lett. A 172, 6 (1992) K. Berndl
  et al., Il Nuovo Cimento Vol. 110 B, N. 5-6 (1995).

  Peter Holland's book The Quantum Theory of Motion (Cambridge
  University Press 1993) contains many pictures of numerical simulations
  of Bohmian trajectories.

  8.  Lawrence R. Mead [e-mail] The Transactional Interpre-
  tation of Quantum Mechanics

  The transactional interpretation of quantum mechanics (J.G. Cramer,
  Phys. Rev. D 22, 362 (1980) ) has received little attention over the
  one and one half decades since its conception. It is to be emphasized
  that, like the Many-Worlds and other interpretations, the
  transactional interpretation (TI) makes no new physical predictions;
  it merely reinterprets the physical content of the very same
  mathematical formalism as used in the ``standard'' textbooks, or by
  all other interpretations.

  The following summarizes the TI. Consider a two-body system (there are
  no additional complications arising in the many-body case); the
  quantum mechanical object located at space-time point (R_1,T_1) and
  another with which it will interact at (R_2,T_2). A quantum mechanical
  process governed by E=h0, conservation laws, etc., occurs between the
  two in the following way.

  1) The ``emitter'' (E) at (R_1,T_1) emits a retarded ``offer wave''
  (OW) \Psi.  This wave (or state vector) is an actual physical wave and
  not (as in the Copenhagen interpretation) just a ``probability'' wave.

  2) The ``absorber'' (A) at (R_2,T_2) receives the OW and is stimulated
  to emit an advanced ``echo'' or ``confirmation wave'' (CW)
  proportional to \Psi at R_2 backward in time; the proportionality
  factor is \Psi* (R_2,T_2).

  3) The advanced wave which arrives at 'E' is \Psi \Psi* and is
  presumed to be the probability, P, that the transaction is complete
  (ie., that an interaction has taken place).

  4) The exchange of OW's and CW's continues until a net exchange of
  energy and other conserved quantities occurs dictated by the quantum
  boundary conditions of the system, at which point the ``transaction''
  is complete. In effect, a standing wave in space-time is set up
  between 'E' and 'A', consistent with conservation of energy and
  momentum (and angular momentum). The formation of this superposition
  of advanced and retarded waves is the equivalent to the Copenhagen
  ``collapse of the state vector''. An observer perceives only the
  completed transaction, however, which he would interpret as a single,
  retarded wave (photon, for example) traveling from 'E' to 'A'.

  Q1. When does the ``collapse'' occur?

  A1. This is no longer a meaningful question. The quantum measurement
  process happens ``when'' the transaction (OW sent - CW received -
  standing wave formed with probability \Psi \Psi*) is finished - and
  this happens over a space-time interval; thus, one cannot point to a
  time of collapse, only to an interval of collapse (consistent with
  relativity).

  Q2. Wait a moment. What you are describing is time reversal invariant.
  But for a massive particle you have to use the Schrodinger equation
  and if \Psi is a solution (OW), then \Psi* is not a solution. What
  gives?

  A2. Remember that the CW must be time-reversed, and in general must be
  relativistically invariant; ie., a solution of the Dirac equation.
  Now (eg., see Bjorken and Drell, Relativistic QM), the nonrelativistic
  limit of that is not just the Schrodinger equation, but two
  Schrodinger equations: the time forward equation satisfied by \Psi,
  and the time reversed Schrodinger equation (which has i --> -i) for
  which \Psi* is the correct solution. Thus, \Psi* is the correct CW for
  \Psi as the OW.

  Q3. What about other objects in other places?

  A3. The whole process is three dimensional (space). The retarded OW is
  sent in all spatial directions. Other objects receiving the OW are
  sending back their own CW advanced waves to 'E' also. Suppose the
  receivers are labeled 1 and 2, with corresponding energy changes E_1
  and E_2. Then the state vector of the system could be written as a
  superposition of waves in the standard fashion. In particular, two
  possible transactions could form: exchange of energy E_1 with
  probability P_1=\Psi_1 \Psi_1*, or E_2 with probability P_2=\Psi_2
  \Psi_2*. Here, the conjugated waves are the advanced waves evaluated
  at the position of R_1 or R_2 respectively according to rule 3 above.

  Q4. Involving as it does an entire space-time interval, isn't this a
  nonlocal ``theory''?

  A4. Yes, indeed; it was explicitly designed that way. As you know from
  Bell's theorem, no ``theory'' can agree with quantum mechanics unless
  it is nonlocal in character. In effect, the TI is a hidden variables
  theory as it postulates a real waves traveling in space-time.

  Q5. What happens to OW's that are not ``absorbed'' ?

  A5. Inasmuch as they do not stimulate a responsive CW, they just
  continue to travel onward until they do. This does not present any
  problems since in that case no energy or momentum or any other
  physical observable is transferred.

  Q6. How about all of the standard measurement thought experiments like
  the EPR, Schrodinger's cat, Wigner's friend, and Renninger's negative-
  result experiment?

  A6. The interpretational difficulties with the latter three are due to
  the necessity of deciding when the Copenhagen state reduction occurs.
  As we saw above, in the TI there is no specific time when the
  transaction is complete. The EPR is a completeness argument requiring
  objective reality.  The TI supplies this as well; the OW and CW are
  real waves, not waves of probability.

  Q7. I am curious about more technical details. Can you give a further
  reference?

  A7. If you understand the theory of ``advanced'' and ``retarded''
  waves (out of electromagnetism and optics), many of the details of TI
  calculations can be found in: Reviews of Modern Physics, Vol. 58, July
  1986, pp. 647-687 available on the WWW as:
  http://mist.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html

  9.  Complex probabilities

  References; Saul Youssef Quantum Mechanics as Complex Probability
  Theory, hep-th 9307019.  S. Youssef, Mod.Phys.Lett.A 28(1994)2571.

  10.  Quantum logic

  References: R.I.G. Hughes, The Structure and Interpretation of Quantum
  Mechanics, pp. 178-217, Harvard University Press, 1989.

  11.  Consistent histories

  References: R. B. Griffiths, Consistent Histories and the
  Interpretation of Quantum Mechanics, Journal of statistical Physics.,
  36(12):219-272(1984)

  M. Gell-Mann and J. B. Hartle, in Complexity, Entropy and the Physics
  of Information, edited by W. Zurek, Santa Fe Institute Studies in the
  Sciences of Complexity Vol. VIII, Addison-Wesley, Reading, 1990. Also
  in Proceedings of the $3$rd International Symposion on the Foundations
  of Quantum Mechanics in the Light of New Technology, edited by S.
  Kobayashi, H. Ezawa, Y. Murayama and S. Nomura, Physical Society of
  Japan, Tokyo, 1990

  R. B. Griffiths, Phys. Rev. Lett. 70, 2201 (1993)

  R. Omn`es, Rev. Mod. Phys. 64, 339 (1992)

  In this approach serious problems arise. This is best pointed out in:
  B. d'Espagnat, J. Stat. Phys. 56, 747 (1989)

  F. Dowker und A. Kent, On the Consistent Histories Approach to Quantum
  Mechanics, University of Cambridge Preprint DAMTP/94-48, Isaac Newton
  Institute for Mathematical Sciences Preprint NI 94006, August 1994.

  12.  Spontaneous reduction models

  Reference:

  G. C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34, 470 (1986).

  13.  What is needed?

  All comments suggested and contributions are welcome. We currently
  have nothing but references on Complex Probabilities, Quantum Logic,
  Consistent Histories and Spontaneous Reduction Models. The entries on
  the following topics are minimal and should be replaced by complete
  articles.

  o  Copenhagen interpretation

  o  Relative State (Everett)

  o  Shut up and calculate

  o  Bohm's theory

  Alternative views on any of the topics and suggestions for additional
  topics are welcome.

  14.  Is this a real FAQ?

  Paul Budnik [e-mail]

  A FAQ is generally understood to be a reasonably objective set of
  answers to frequently asked questions in a news group. In cases where
  an issue is controversial the FAQ should include all credible opinions
  and/or the consensus view of the news group.

  Establishing factual accuracy is not easy. No consensus is possible on
  interpretations of QM because many aspects of interpretations involve
  metaphysical questions. My intention is that this be an objective
  accurate FAQ that allows for the expression of all credible relevant
  opinions.  I did not call it a FAQ until I had significant feedback
  from the `sci.physics' group. I have responded to all criticism and
  have made some corrections. Nonetheless there have been a couple of
  complaints about this not being a real FAQ and there is one issue that
  has not been resolved.

  If anyone thinks there are technical errors in the FAQ please say what
  you think the errors are. I will either fix the problem or try to
  reach on a consensus with the help of the `sci.physics' group about
  what is factually accurate.  I do not feel this FAQ should be limited
  to noncontroversial issues.  A FAQ on measurement in quantum mechanics
  should highlight and underscore the conceptual issues and problems in
  the theory.

  The one area that has been discussed and not resolved is the status of
  locality in Everett's interpretation. Here is what I believe the facts
  are.

  Eberhard proved that any theory that reproduces the predictions of QM
  is nonlocal1. This proof assumes contrafactual definiteness (CFD) or
  that one could have done a different experiment and have gotten a
  definite result. This assumption is widely used in statistical
  arguments.  Here is what Eberhard means by nonlocal:

     Let us consider two measuring apparata located in two different
     places A and B. There is a knob a on apparatus A and a knob b on
     apparatus B.  Since A and B are separated in space, it is
     natural to think what will happen at A is independent of the
     setting of knob b and vice versa.    The principles of relativity
     seem to impose this point of view if the time at which the knobs
     are set and the time of the measurements are so close that, in
     the time laps, no light signal can travel from A to B and vice
     versa. Then, no signal can inform a measurement apparatus of
     what the knob setting on the other is. However, there are cases
     in which the predictions of quantum theory make that
     independence assumption impossible. If quantum theory is true,
     there are cases in which the results of the measurements A will
     depend on the setting of the knob b and/or the results of the
     measurements in B will depend on the setting of the knob a.1

  It is logically possible to deny CFD and thus to avoid Eberhard's
  proof.  This assumption can be made in Everett's interpretation.
  Everett's interpretation does not imply CFD is false and CFD can be
  assumed false in other interpretations.  I do not think it is
  reasonable to deny CFD in some experiments and not others but that is
  a judgment call on which intelligent people can differ.

  It is mathematically impossible to have a unitary relativistic wave
  function from which one can compute probabilities that will violate
  Bell's inequality. A unitary wave function does satisfy CFD and thus
  is subject to Eberhard's proof. This is a problem for some advocates
  of Everett who insist that only the wave function exists.  There is no
  wave function consistent with both quantum mechanics and relativity
  and it is mathematically impossible to construct such a function.
  Quantum field theory requires a nonlocal and thus nonrelativistic
  state model. The predications of quantum field theory are the same in
  any frame of reference but the mechanisms that generate nonlocal
  effects must operate in an absolute frame of reference. Quantum
  uncertainty makes this seemingly paradoxical situation possible. There
  is a nonlocal effect but we cannot tell if the effect went from A to B
  or B to A because of quantum uncertainty. As a result the predictions
  are the same in any frame of reference but any mechanism that produces
  these predictions must be tied to an absolute frame of reference.

  There is a certain Alice in Wonderland quality to arguments on these
  issues. Many physicists claim that classical mathematics does not
  apply to some aspects of quantum mechanics, yet there is no other
  mathematics. The wave function model is a classical causal
  deterministic model. The computation of probabilities from that model
  is as well.  The aspect of quantum mechanics that one can claim lies
  outside of classical mathematics is the interpretation of those
  probabilities.  Most physicists believe these probabilities are
  irreducible, i. e., do not come from a more fundamental deterministic
  process the way probabilities do in classical physics. Because there
  is no mathematical theory of irreducible probabilities one can invent
  new metaphysics to interpret these probabilities and here is where the
  problems and confusion rest.    Some physicists claim there is new
  metaphysics and within this metaphysics quantum mechanics is local.

  References:

  P. H. Eberhard, Bell's Theorem without Hidden Variables, Il Nuovo
  Cimento, V38 B 1, p 75, Mar 1977.