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Bell's theorem states that any physical theory that incorporates local realism and the no-conspiracy assumption cannot reproduce all the predictions of quantum mechanical theory.

Hence, we cannot develop a theory of local hidden variables if we assume non-conspiracy.

If we assume the freedom of choice of the experimenter (to, for example, measure the x-axis rather than the y-axis in an experiment), how do we maintain determinism in, for example, the MWI? Doesn't the very fact that she has freedom to choose necessitate that her actions are not physically determined?

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Bonj
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    You are talking philosophy on a physics site. It doesn't belong here. Words like "local realism", "non-conspiracy", "freedom to choose" do not have testable physical ontologies. There are no experiments that would allow us to pull back the curtain and unmask the man behind it that the quantum mystics have been looking for all these decades. Truthfully, there is no curtain, there is only a serious misunderstanding of the world here. The world is quantum mechanical and classical mechanics follows in the weak measurement limit, which only works one way. There is nothing "the other way around". – CuriousOne May 13 '16 at 01:00
  • But that's just an easy answer. A few years ago, physicists used to like to explain their mathematical formalism, since it is clearly just one aspect of a theory. And science is all about theories, that's why we have interpretations. – Bonj May 13 '16 at 16:39
  • It's an easy answer because it is the only answer. The things you mention are not part of physics. They don't belong into quantum mechanics because you don't need any of them to explain even a single datapoint that has been gathered in the last 100+ years. Science is not about theories. It's about observations and what you got there is not even theory. – CuriousOne May 13 '16 at 20:23

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The assumption of "freedom" in the Bell inequality derivation is a somewhat misleading statement. It is not an assumption of freedom in the philosophic sense. Rather, it is the statement that the experimenter's choice of measurement basis should not be correlated with the quantum state they are measuring.

For example, one could set the measurement apparatus such that for each run of the experiment the measurement axis is determined by the last digit of the Dow Jones at that moment. Then the relevant assumption of the Bell inequality derivation is that the laws of nature not somehow determine the stock market so that it fluctuates in sync with the secret changes in the true orientation of the entangled spins. You can see why theories that break this assumption are often described as "conspiratorial." You can also see that science itself depends on making assumptions like this. You can't trust the result of any experiment if you start to suppose that, say, the laws of nature are set up so that when you do a particular experiment you always suffer a hallucination that makes you think that you got the opposite result. This is why most people tend to dismiss this possibility. That said, some people do try to construct theories that violate this assumption without being as egregiously conspiratorial as the above example- see, e.g., this recent question.

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Rococo
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  • None of this is even borderline necessary. Nature tells us very clearly how quantum systems evolve and what we can expect from measurements. She has also told us very clearly that she is not a classical machine. There are no internal contradictions in our theory until someone introduces the notion that she has to be classical, after all, and that we have to force this unnecessary assumption onto the theory. The Bell inequality, for instance, doesn't make a scientific statement about nature. It makes a meta-statement about our expectations about nature. That's not science but philosophy. – CuriousOne May 13 '16 at 20:25
  • @CuriousOne While you are entitled to such an opinion, grounded in your personal philosophy about what is and is not a valid scientific inquiry, I find comments like this the very definition of "non-constructive," and will start flagging them accordingly. – Rococo May 13 '16 at 23:19
  • It's not my personal philosophy but the definition of the scientific method, which is about the rational explanation of observed phenomena. Physics, in particular, requires us to reduce complex observed phenomena to simple observed phenomena (symmetries, conservation laws etc.). It does not compel us to ask questions that nature doesn't pose and nature has certainly never asked us to shoehorn classical mechanisms into quantum mechanics by all means possible. Quite the contrary, the natural observation is that CM emerges from QM, although it doesn't do so the way it's often taught in QM101. – CuriousOne May 13 '16 at 23:24
  • @CuriousOne I think you put too much emphasis on the comparison with classical physics. I tend to think of Bell's theorem and associated results as providing a pithy statement of how much correlation is possible between observables in QM (or a general class of theories including QM). Such a rule certainly fits your definition of science. Since this isn't really a suitable venue for this discussion I will leave it at that. – Rococo May 13 '16 at 23:59
  • @Rococo As I see it, these comments are suggestions for improvement of your answer (in a manner of speaking), which qualifies them as constructive. – David Z May 14 '16 at 07:46
  • "The assumption of "freedom" in the Bell inequality derivation...is the statement that the experimenter's choice of measurement basis should not be correlated with the quantum state they are measuring."

    I understand you, but I'm still a little puzzled. Suppose that the experimenter's choice was indeed correlated with the quantum state. What would this imply?

    Furthermore, to say that, in your example, the stock market is the cause of the choice is surely incorrect. In reality, the experimenters choice to rely on the stock market is the cause.

     – Bonj May 14 '16 at 18:03
  • (1/2) @Bonj This is true. One must assume that every part of the logical chain is uncorrelated with the state: the stock market, the choice of the experimenter to use the stock market, and the reliability of the actual hardware used to implement the experiment. Ultimately you will have to decide how plausible this seems for yourself. – Rococo May 14 '16 at 18:41
  • (2/2) But I would again note that it seems to me that assumptions of this sort, i.e. that it is at least possible in principle to make a measurement that does not depend on such details of the measurer, underlie the very notion of learning things about the objective world by experiment. – Rococo May 14 '16 at 18:41
  • Thanks, but did you answer the first question? What would it mean for the correlation to be present? Would that mean that the experimenter is willfully determining the state? Or that the state is "willfully" determining the variable of the experimenter? I know this sounds airy now. – Bonj May 14 '16 at 21:34
  • @Bonj I intentionally did not address that question because I don't think I can give a explanation that is more than a literal rephasing of the statement: the properties of the physical world are inherently correlated with (so either determining, or determined by, or both determined by the some other thing) the measurement choices. As I mentioned in the main answer, there is some disagreement about how "bad" this is- I suggest you search for "superdeterminism" on this site and google, and you'll see several nice discussions from different points of view. – Rococo May 15 '16 at 00:16
  • Such as http://physics.stackexchange.com/questions/106725/why-is-superdeterminism-generally-regarded-as-a-joke ,

    http://backreaction.blogspot.com/2013/10/testing-conspiracy-theories.html

    http://motls.blogspot.com/2013/10/superdeterminism-ultimate-conspiracy.html

    undoubtedly many others too

     – Rococo May 15 '16 at 00:17