# Conway & Kochen's Free Will Theorem (2006) Free Will Implies Quantum Indeterminacy

The freedom of an experimenter to set up an experiment as he wishes implies that certain quantum systems have indeterminate measurement outcomes. So claim Conway and Kochen, in this paper . Their views are radical. They see quantum indeterminacy as essentially the same thing as free will, but writ small. Every time the wave packet collapses, they regard it as a little act of free will. And human free will they postulate to be the "integrated up" consequence of many contributing, quantum mechanically indeterminate, microscopic events.

The core of the theorem is essentially similar to that of the famous Kocken & Specker paradox, J.Maths & Mech 17 (1967) 59-87. The version in the 2006 paper of Conway & Kochen is a specific realisation of the general case. However, it is all the more convincing for being a concrete illustration. These theorems show that the outcomes of certain measurements are simply not compatible with these same outcomes being pre-determined.

The particular example in the 2006 paper concerns a spin 1 particle. It is possible to measure the square of the spin component on each of three mutually perpendicular directions, x, y and z. (This is possible because the squares of the spin component operators commute for a spin 1 particle, even though the component operators themselves do not). Moreover, the standard formulation of quantum mechanical spin shows that two of these measurements will produce the result 1, and the third will be 0, though the order is unknown in general.

Now suppose that the results of any such measurement are determined beforehand. Then, for any perpendicular directions, x, y, z, exactly two produce '1' and the third produces '0'. If the outcome of these spin measurements is determined in advance, this means that we can consider a sphere and associate with every point on the surface of the sphere either the number 1 or the number 0, these being the postulated outcomes of the measurement in that direction. But we require that for every triple of mutually perpendicular directions we may choose, exactly two are '1' and the third '0'. [Moreover, because we are measuring spin-squared, diametrally opposite points are equal].

But it can easily be established that no such assignment of 1s and 0s to the surface of a sphere can have this property (see Conway and Kochen for a proof).

The important thing to appreciate is that the quantum mechanical prediction, namely that the three outcomes are always {1,1,0} in some order, is experimentally verified. The specific formalism of quantum mechanics is not strictly pertinent to the issue, as Conway and Kochen stress. As long as it is established experimentally that the outcomes are always {1,1,0} in some order, then it follows that the results of measurements on this system cannot be determinate. The only way out of a flat contradiction is that the result of the spin measurement in, say, the x direction, can be different if the other directions, y and z, were chosen differently, e.g. to be y' and z' instead.

Ah! This is the answer, then, is it? Determinism can be preserved if an interaction occurs due to the setting up of the apparatus in a different orientation which "tips off" the x-direction measurement as to which outcome it needs to produce. But, no, this loophole can be closed by virtue of another feature of quantum mechanics: entanglement.

This goes back to the EPR paradox. A pair of spin 1 particles can be produced such that the outcomes of measurements are the same on both, even if the two particles are causally separated (e.g. by being a long way apart). We decide that we shall make a measurement only in the x-direction on the second particle, not in the y or z directions. So, assuming causal separation of the two particles, the x-direction measurement on the second particle cannot be influenced by the orientation of y and z used to measure spins on the first particle. But since the results of the x-direction spins are necessarily the same for both particles, it follows that the x-direction measurement on the first particle cannot, after all, be influenced by the y and z orientations. So that attempt at an explanation fails.

Again, it is worth stressing that the crucial entanglement property, in this case simply the equality of certain measurement outcomes for a certain bipartite state (namely the singlet state of the composite system) has been tested experimentally. At which point, the formal machinery of quantum mechanics is no longer necessary to predict this effect - we can merely take it as a bald experiment fact - as do Conway & Kochen.

So there we have it: quantum indeterminacy is proved. Hidden variables are dead. Except that there is one last snag. We have assumed in the above argument that the experimenter is capable of making a free and arbitrary choice regarding the orientation of the measurement axes. But what if this is not the case? Perhaps the experimenter is unwittingly an automaton, always conforming in his "decision" to the choices which are consistent with a predetermined set of spin outcomes. Any triple of points on the sphere which do not conform to the {1,1,0} rule, as some must, is simply impossible for our robot experimentalist to choose. In other words, the last obstacle in the way of proving quantum indeterminacy is whether we can be sure that the experimenter has free will. The existence of free will is the only extra ingredient which must be postulated in order for quantum indeterminacy to be unavoidable.

The logical inverse of this, that quantum indeterminacy implies free will, is not proved by the theorem. Nor does the theorem prove that free will and quantum indeterminacy are essential, only that the former implies the latter. A deterministic universe in which free will does not exist but instead our every action is dictated by clockwork is not logically ruled out. The situation is analogous to that in regard to solipsism. It is logically possible that nothing exists outside myself, but it is a profoundly unsatisfactory philosophical position. The only sensible position to take in regard to quantum mechanics is that it truly is indeterminate.

Whether quantum indeterminacy really does play an essential role in giving rise to free will, as Conway and Kochen believe, is another matter entirely. I am inclined to think not. If some source of indeterminacy is required to produce free will, then I am not convinced that it must be true quantum indeterminacy. A functionally equivalent degree of randomness can be simulated by purely deterministic means, as for example is the case for random number generators supplied with computing codes. Providing that the underlying deterministic mechanics is sufficiently complicated, and preferably unrelated to the subject at hand, then surely some internal roulette wheel spinning, or coin tossing, will meet any such needs?

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