Scott. You seem like a smart guy who knows his way around QM, so I have a few questions about the “Bell’s theorem” proof.

Bell main criticism of von Neumann’s no-go theorem is as follows:

“The essential assumption can be criticized as follows. At first sight the required additivity of expectation values seems very reasonable, and it is rather the nonadditivity of allowed values (eigenvalues) which requires explanation. Of course the explanation is well known: A measurement of a sum of noncommuting observables cannot be made by combining trivially the results of separate observations on the two terms — it requires a quite distinct experiment.”

Yet in every “proof” of Bell’s theorem I’ve come across, expectation values from QM are simply combined linearly in an inequality expression (which is valid BTW) to claim violation. So when Bell wrote in is argument against von Neumann that:

“It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made.”

Why is this not also a criticism of Bell’s own theorem? How can Bell’s theorem be valid if the proof relies on a linear combination of expectation values, of incompatible measurements contrary to the principles of QM?

]]>To settle this, would you accept an explicit demonstration that sonon quasiparticles have spin-half symmetry and behave precisely like the quantum mechanical particles analysed in Bell’s original paper, including violating Bell’s inequality?

]]>**Scott** posts “But as soon as you ask for the “state” of the system — i.e., for an object sufficient to probabilistically predict the outcome of any possible measurement that could be made in the future — the exponential character of Hilbert space comes roaring back.”

Unless the dynamical system couples to a continuum of vacuum states, or (equivalently?) a thermal bath, or (equivalently??) is a product-state pullback. For some reason (**yet what might that reason be**?) Nature *requires* that both her external reality and human laboratory experiments respect these coupling-to-continuum constraints. That’s why it’s been heartening in recent years (for us system engineers) to witness the gradual weakening of theoretical faith in the absolute reality of unitary evolution on finite-dimensional Hilbert spaces!

- I would like to ask you (and to the readers) your idea about the fact that BQP is a subset of PSPACE. It seems that something huge is needed, but not necessarily a “large Hilbert space”: a very long calculation time can do the work. Do you think that this can say something fundamental on what QM is? Not an easy equation in a huge Hilbert space but an extremely difficult problem in a smaller space?

Actually yes, I’ve been telling people for a while that BQP⊆PSPACE is a deep and underappreciated fact about the foundations of quantum mechanics! (One of my laugh lines is that Feynman won the Nobel Prize in physics basically for pointing out that BQP⊆P^{#P}⊆PSPACE—i.e., that you can organize QFT calculations as a giant sum rather than keeping a whole wavefunction in memory.)

On the other hand, I don’t see this as a challenge to the Hilbert space formalism, but as a *property* of the formalism: a property of “modesty,” if you like. We never observe a naked state vector in the wild; we only ever observe the outcomes of measurements. And if you only care about predicting the outcomes of measurements specified in advance, you can ditch the notion of “states” almost entirely, and organize your calculations in a more efficient way (*just how much* more efficient being an active research topic). But as soon as you ask for the “state” of the system — i.e., for an object sufficient to probabilistically predict the outcome of any *possible* measurement that could be made in the future — the exponential character of Hilbert space comes roaring back.

*YAWN…* next!

Yes, Euler’s equation contains only local interactions. Nevertheless, the energy and angular momentum of a vortex are delocalised in the fluid. I believe this means a vortex has at least some properties which are not localised at the core. A sonon has the same delocalised properties.

Would you be convinced by an explicit proof of the spin correlation in Bell’s original paper (which he shows violates his inequality)?

I am afraid I can’t quantify how ‘hard’ it would be to break the current experimental glass ceiling. You would have to get a single particle to lose coherence with system A, fall into coherence with B, revert to A and so on, with the net effect that it remains in coherence with both. I think this would be exceedingly difficult, but it is at least mathematically conceivable. If someone can achieve it they might be able to break the glass ceiling. If it’s not clear why, the presentation (on my web site) might be helpful.

]]>Scientists discover a way around Heisenberg’s Uncertainty Principle

According to a pair of scientists from the University of Rochester and the University of Ottawa, there may be a way around Heisenberg’s famous Uncertainty Principle.

According to a report published this week in Nature Photonics, a recently developed technique that allows scientists to directly measure the polarization states of light could be the key. The direct measurement technique, developed in 2011, allows scientists to measure the wavefunction – a way of determining the state of a quantum system.

The pair of scientists say the new technique relies on a “trick” that measure the first property of a system, leaving the remaining parties untouched. The careful measurement relies on the “weak measurement” of the first property followed by a “strong measurement” of the second property, the pair writes in the report.

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