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Dear Ken,

Thank you for your comments, but I think you may have misunderstood my point. When Alice says, on the basis of the output of her apparatus, that the particle had the property S_x = -1/2 when it entered her detector, she is NOT referring to a “hidden variable” in the sense that term is commonly used these days, as something in addition to or in place of the quantum Hilbert space. Instead, S_x = -1/2 refers to the quantum property represented by the subspace of the spin-half Hilbert space that contains the ket |x-> and whose projector is

[x-] =|x-><x-|. Well-designed measurements measure quantum properties, and what I call the second measurement problem is to show, using quantum mechanics without ‘measurement’ as an axiom, that Alice’s conclusion is correct. See my ‘Consistent Histories Essentials’ in this workshop for additional comments; in particular my “EPR, Bell, and quantum locality”, Am. J. Phys., 79:954–965, 2011, arXiv:1007.4281 gives a lot of details.

Regarding your use of statistical independence and its violation in the case of hidden variables as evidence for retrocausality, let me respond as follows. First, one can construct a perfectly consistent probability theory for quantum mechanics using Hilbert subspaces and taking account of the fact that the projectors for different properties in general do not commute: in the spin half case [x-] and [z+] do not commute, so you cannot put both of them into the same sample space. Among other things this gets rid of any notion of mysterious nonlocal influences in the EPR-Bohm situation–more on that in my reply to Mark Stuckey–thus undermining one reason you think you need you retrocausality.

An important consequence of taking proper account of the noncommutation of quantum projectors is in blocking the following paradox: Alice measures S_x and determines an earlier values, say S_x = -1/2. But she could instead have measured S_z and then would have obtained a value for this at the earlier time. Therefore a spin-half particle has both an S_x and an S_z value at the same time. (Doesn’t this sound a bit like EPR?) It is clear that such cannot be the case in Hilbert space quantum mechanics, since there is no ray in the Hilbert space that combines the two incompatible properties. And it supplies the real reason behind the textbook assertion that one cannot simultaneously measure both S_x and S_z for a spin-half particle: what does not exist cannot be measured!

Alice’s choice of which measurement to perform is best regarded as determining the TYPE of information she acquires about the earlier state of the spin-half particle, rather than (retro) causing it to have some property at this earlier time. The trouble with your assertion that Alice “always infers a prior spin that is aligned with her choice of measurement (or anti-aligned)” is that it is based upon a common but inadequate way of visualizing a spin-half particle. We tend to think of such things as a little gyroscope with a precisely aligned spin axis. This is misleading from a quantum perspective because the gyroscope has zero angular momentum components in directions perpendicular to its spin axis, whereas we know that the sum of the squares in the quantum case is 3/4 hbar^2, not 1/4 hbar^2. While any classical picture will mislead to some extent, I think the following is preferable, and it is what I recommend to my students. Think of the gyroscope axis as randomly aligned, and of |z+> as a random alignment except that the z component of its angular momentum is positive. Alice can only measure one component of angular momentum, so her answer will depend upon which component she chooses to measure, but the value (for that component) is already there in advance. As I say, one has to treat such classical analogies cautiously; they are not a substitute for a proper analysis based on Hilbert subspaces, but they do help to provide some intuitive feel for what can otherwise be rather abstract mathematics. When classical analogies are taken too seriously one ends up with various hidden variable oddities, such as nonlocal influences, and contextualy of quantum measurements, which a proper quantum analysis shows are incorrect.

In summary, I do not think retrocausality is needed to understand EPR-Bohm. Things are local, causal, and noncontextual, provided you use Hilbert subspaces rather than (classical) hidden variables.

Bob Griffiths

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