Physics Dopers: "Twinning" behavior?

[Frylock]

On page 25 of this paper (It's Conway's first paper on his Free Will Theorem) he writes:

Although we find ourselves unable to give an
operational definition of either “free” or “random,” we have managed to distinguish
between them in our context, because free behavior can be twinned,
while random behavior cannot (a remark that might also interest some philosphers
of free will).

I have tried to read through the section of the paper that explains what it means to "twin" behavior, but it's over my head.

Can anyone here explain it to a layman?

[Half Man Half Wit]

It basically refers to quantum entanglement -- Alice getting a spin up result in her measurement on one part of an entangled particle pair necessitates Bob's getting a spin down if he measures along the same axis, instantaneously, regardless of spatial separation. I'm not exactly sure how this applies to 'free behaviour', though, but I can see how it might be meant that it's impossible for 'random' behaviour to be twinned, since if one of the partners behaves randomly, the other can't, due to their entanglement -- the same way Alice's result is completely random, and Bob's then completely determined.

[ZenBeam]

Originally Posted by article

It is possible to produce two distantly separated spin 1 particles that are
“twinned,” meaning that they give the same answers to corresponding questions
2. A symmetrical form of the TWIN axiom would say that if the same
triple x, y, z were measured for each particle, possibly in different orders, then
the two particles’ responses to the experiments in individual directions would
be the same. For instance, if measurements in the order x, y, z for one particle
produced x → 1, y → 0, z → 1, then measurements in the order y, z, x for the
second particle would produce y → 0, z → 1, x → 1.‡ Although we could use

‡For simplicity, we have spoken of measuring x, y, z in that order, but nothing in the proof
is affected if they are measured simultaneously, as in the “spin-Hamiltonian” experiment of
Endnote 1.

This sounds like entangled particles(*), except that as I understand it, once one of the directions is measured, the entanglement breaks. So while the first particle measurement gives X → 1 (for both particles), the measurements for Y and Z are no longer entangled. In short, I guess, I question their assertion that twinned particles, as defined, exist.

* I've always heard that entangled particles have opposite spin, but that distinction probably isn't important here.

[Chronos]

* I've always heard that entangled particles have opposite spin, but that distinction probably isn't important here.

The easiest ways of producing entangled particles gets you opposite spins, but there probably is some way to get them with the same. You're right, though, that it isn't an important distinction.

[Omphaloskeptic]

Originally Posted by ZenBeam

This sounds like entangled particles(*), except that as I understand it, once one of the directions is measured, the entanglement breaks. So while the first particle measurement gives X → 1 (for both particles), the measurements for Y and Z are no longer entangled. In short, I guess, I question their assertion that twinned particles, as defined, exist.

* I've always heard that entangled particles have opposite spin, but that distinction probably isn't important here.

It's a little more complicated than this. What you say is correct for the common example of a pair of spin-1/2 particles in the "spin-0 (singlet) state" |↑↓>-|↓↑> (*). A measurement of either particle's spin along the z axis will project the state into either |↑↓> or |↓↑>, so the two particles' spins must be oppositely directed. It turns out that the spin-0 state is rotationally invariant, so the same result actually holds for measurements along any axis.

However, the example given in the paper is for two spin-1 particles. These live in a three-dimensional Hilbert space, not a two-dimensional space like the spin-1/2 particles. This means that measuring the spin of one of these particles along the z axis has three possible results: |↑> (spin +1), |↓> (spin -1), and |0> (spin 0). The measurement proposed in the paper is chosen so that it does not distinguish between the states |↑> and |↓> (this is what it means when they say that it is measuring "the square of the component spin"); so a measurement result of 0 projects onto the state |0>, but a measurement result of 1 projects onto the subspace spanned by the states |↑> and |↓>. It turns out (as a simple consequence of the quantum-mechanics angular-momentum rules) that for a spin-1 particle, measuring the squares of the x, y, and z components of spins always produces two 1s and one 0. This is their "SPIN axiom" (discussed in their Endnote 1).

Because a single measurement does not fully project the particle state, it does not fully destroy the entanglement. This is what allows the three measurements to all give correlated results. As discussed in Endnote 2, the two-particle state is again the singlet state, a rotationally-invariant state of total spin 0. If the two experimenters actually measured the z-components of spin, they would get opposite results (|↑> and |↓> or |0> and |0>), but since they are only measuring the squared z-component, they always get the same answer.

(*) I'm ignoring normalization.