Saturday, April 29, 2017

Quantum measurement and Free Will

John Bell invoked Free Will for experimenters as part of a derivation of inequalities that would have to be satisfied by classical relativistic models\cite[Bell, Chapter 12], a modification of an earlier stipulation that experimental choices should be "at the whim of experimenters"[Bell, Chapter 7].
He more pragmatically requires that the experimenters' choices should be ``effectively free for the purpose at hand'', which suggests some consideration of just how free that might be in the context of quantum measurement.

Consider Alice and Bob running two ends of an experiment. Alice and Bob each have to choose a random sequence of 0s and 1s. If either of them chooses 0 too often or 1 too often, we have to restart the data collection. They're also not allowed to have too many 0000 sequences, too many 01101110 sequences, et cetera; they have to satisfy all the tests here, say, within some pre-agreed limits. They're not allowed to look at the statistics of their past choices to make sure that they don't break any of the rules. A typical experiment might need Alice and Bob each to generate a sequence that contains a few hundred million 0s and 1s that can be certified after the event to be random enough. Furthermore, without conferring, the two lists must not be correlated, again within some pre-agreed limit. Hard to do. Alice and Bob don't seem to be very free at all. Every individual 0 or 1 can be freely chosen, but the statistics are constrained.

Alice and Bob in practice farm out the job either to random number generators or to photon detectors driven by light from stars 600 light years away (arXiv here). No Free Will required.

I don't have much problem with Bell-EPR experiments these days, but the seemingly pervasive idea that Free Will plays a significant part in the discussion seems unsupportable.

The discussion above hints at the stochastic nature of the constraints on Free Will. Suppose that Alice and Bob are both friends of Wigner. They agree that Wigner can construct quantum mechanical models of their brains that predict the statistics of their choices, which is checked while they practice choosing a list that contains millions of random numbers, millions of times. If quantum theory is truly universal, this is just hard to do, even very hard, but it's not in principle impossible. This model doesn't constrain Alice and Bob's Free Will, it just describes where their Free Will has brought them to. If Alice and Bob include observations of stars 600 light years away to decide their 0 and 1 choices, then Wigner has to include a quantum mechanical model of the light from those stars that is accurate enough to describe the statistics of Alice's and Bob's lists. A quantum mechanical model describes the statistics of Alice's and Bob's choices about as much as would a classical stochastic model.

Bell J S 1987 Speakable and unspeakable in quantum mechanics (Cambridge: Cambridge University Press).

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