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Two observers A and B measure a quantum entangled state and obtain correlated results, even if their separation is space-like (each is out of the light cone of the other). A possible interpretation is that the observer who makes the first measurement (say A) collapses the quantum state, thus fixing the result of the other observer's (B) measurement. But there are frames of references in which B's measurement comes first. This seems a paradox. What is the solution?

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The solution is that the wave function isn't any real wave; the wave function is a tool, a collection of complex numbers that an observer may use to predict the results of the experiments.

And the collapse of the wave function isn't a real process in which a material object changes its shape; it's only the process in which an observer learns some facts about the physical system, so he has to replace all overall probabilities – which described what he knew before – by the conditional probabilities in which the condition "measured properties are exactly what was measured" is taken into account as an assumption.

When one carefully separates things that are physical and that genuinely exist – results of experiments that may be predicted – from the fantasies that one may want to "imagine to operate inside", it is very clear that none of the two reference frames plays a more important role than the order and that Nature never has to solve the problem "which of the measurements occurred first".

In other words, the question "which of the measurements occurred first" or "which of the wave functions collapsed first" doesn't have an objective answer – and indeed, relativity prohibits an objective ordering of spacelike-separated events – but that's fine from the quantum mechanics viewpoint as well because the collapse is indeed a subjective process only, too. So different observers may have a different story about the collapses and their orderings – because the collapses are nothing else than their subjective learning of the new facts.

The actual objective – or at least "intersubjective" and "approximately objective" for all practical purposes – facts, namely the results of measurements and/or their probabilities, are always predicted to be identical regardless of the reference frame to use. And that's the only thing that relativity requires: the objective events have to obey Lorentz-invariant laws. Subjective interpretations and unphysical parts of the story that people are adding to "imagine" what's happening before the experiments may depend on the reference frames because they're totally unphysical.

The correlation between A and B, the two events in which two entangled subsystems are measured, isn't evidence for the influence of A on B or B on A. It's an example of the statement that "correlation doesn't mean causation". Indeed, relativity prohibits both influence of A on B and B on A. Instead, the correlations are the consequence of the action of I on A and I on B where I is the initial state, an event in which the subsystems A and B were in contact before they were separated. The observed correlations don't mean that there is any influence propagating between A and B; there's no such an influence. People's belief that there has to be such an influence is always an artifact of their attempts to imagine a "classical model that emulates quantum mechanics". But the world isn't classical and there's no reason why classical models should lead to the right conclusions, and indeed, when it comes to the question whether A,B are influencing one another, any classical model inevitably produces the wrong conclusion that A,B have to influence one another. Relativity prohibits any such influence and unlike all the "classical models of entanglement", quantum mechanics is fully compatible with this ban.

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Thanks for answering Luboš! I think your knowledge interpretation is nice and clean, but does not explain the experimental fact that the measurements of A and B are actually correlated as quantum mechanics predicts. In a frame of reference where A is the first to measure, say spin up, B will measure spin down. But now switch to a frame of reference where B is the first to measure. How to explain that B measures spin down, in absence of a collapse caused by A measurement (which has not happened yet in this frame of reference)? It seems to me that Everett's MWI is the only way out. –  Giulio Prisco Mar 25 '12 at 17:08
    
Dear Giulio, the correlation is unambiguously and quantitatively predicted by quantum mechanics, in agreement with the experiment, and the quantum calculation makes it self-evident that the reason for the correlation is the contact of the two subsystems in the past (the initial state) and not anything that happens during the measurements. It's nearly guaranteed, both in classical and quantum physics, that two subsystems in contact remain correlated with each other. None of the things provides a glimpse of a rational evidence for MWI, "real collapse", or any of these unphysical superconstructs. –  Luboš Motl Mar 25 '12 at 18:33
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Real measurements take time, and are not instantaneous. To treat the collapse as instantaneous is an idealization, valid for many applications of quantum mechanics.

If relativistic effects play a role, one needs to use quantum field theory. However, the measurement process in quantum field theory is very poorly researched. Thus statements about the conflict of instantaneous collapse and relativity theory are based on very shaky grounds.

For measurement in the relativistic case (but without invoking field theory) in the most down to earth interpretation (i.e., the main stream view), see:

  • A. Peres, Classical interventions in quantum systems. I. The measuring process Phys. Rev. A 61, 022116 (2000). quant-ph/9906023
  • II. Relativistic invariance Phys. Rev. A 61, 022117 (2000). quant-ph/9906034
  • A. Peres, Quantum information and relativity theory Rev. Mod. Phys. 76, 93–123 (2004). quant-ph/0212023

    After having read that, you'll probably be immune against many potential infections in this area. These papers indicate the absence of problems, as far as such a simplified analysis can be trusted.

    A relativistic dynamic collapse model is given in http://lanl.arxiv.org/abs/1003.2774

    [taken from Section ''Is there a relativistic measurement theory?'' in Chapter A4: The interpretation of quantum mechanics of my theoretical physics FAQ.]

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    Dear Dr Neumaier, the measurement in QFT isn't "poorly researched". It's exactly as well researched and understood as measurement in any other quantum mechanical theory because all QM theories share the same postulates and foundations so the explanations are completely universal and hold for all QFTs as well. They only differ in the list of quantities that can be measure, not in the logic how it's measured and what it means. Also, it's a matter of a few lines – known since 1930 or so when QFT was born as a framework – to prove the Lorentz invariance (and thus locality) of quantum field theory. –  Luboš Motl Mar 26 '12 at 4:59
        
    @Lubos: There are thousands of papers on nonrelativistic measurement theory (in a preferred frame) and only few in a covariant setting. Years ago, it took me a long time to even find these. Of course, the results are conclusive. But the research effort is still poor compared to the nonrelativistic case. - How it is measured depends on observers, and these behave differently in relatvistic and in nonrelativistic settings. –  Arnold Neumaier Mar 26 '12 at 8:16
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    "There are thousands of papers on nonrelativistic measurement theory (in a preferred frame) and only few in a covariant setting." - As I explained, it's because there is absolutely no conceptually new mystery associated with the "switch" to relativistic quantum theories. Relativistic quantum theories are just a special case of quantum theories that also preserve some extra symmetry (whose nature has nothing to do with the foundational issues of quantum mechanics); relativistic theories are not qualitatively different from other quantum theories when it comes to the quantum issues. –  Luboš Motl Mar 26 '12 at 11:06
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