Conway's Proof of the Free Will Theorem

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We must believe in free will. We have no choice. -- Isaac B. Singer

Background

In mid-2004, John Conway and Simon Kochen of Princeton University proved the Free-will Theorem. This theorem states "If there exist experimenters with (some) free will, then elementary particles also have (some) free will." In other words, if some experimenters are able to behave in a way that is not completely predetermined, then the behavior of elementary particles is also not a function of their prior history. This is a very strong "no hidden variable" theorem.

This result has not currently been published, however, on January 27, 2005, Dr Conway gave a public lecture at the University of Auckland on the proof of this theorem. The following is my account of his talk. As a disclaimer, please note I am not a physicist and do not understand some of the subtler nuances of quantum mechanics. I have tried to faithfully reproduce Conway's explanation and argument, however, there may still be some errors in this document. If you find such an error, do let me know so I can correct it.

In my (very weak) defense, I quote Dr Conway's retelling of Feynman's quip: If you meet someone who claims to understand quantum mechanics, the only thing you can be sure of is that you have met a liar.

Conway's talk was informative, entertaining and very accessible. The audience consisted not only of mathematicians and physicists — I recognized many computer scientists, philosophers and at least one theologian.

Assumptions

'There exists at least one experimenter in the universe with some free will' implies 'The particles in the universe also have some free will' assuming SPIN, TWIN and FIN.

The Conway-Kochen proof of the Freewill Theorem relies on three axioms they call SPIN, TWIN and FIN:

SPIN
Particles have the 101-property. This means whenever you measure the squared spin of a spin-1 particle in any three mutually perpendicular directions, the measurements will be two 1s and a 0 in some order.
FIN
There is a finite upper bound to the speed at which information can be transmitted.
TWIN
If two particles together have a total angular momentum of 0, then if one particle has an angular momentum of s, the others must necessarily have an angular momentum of -s.

Conway expanded on each of these axioms during his talk. He insisted that given his proof, if you disagreed with his conclusion, you must necessarily also disagree with one of these axioms. These are axioms and so they are stated without proof, however, the two axioms SPIN and TWIN can be experimentally tested and verified. Moreover some of these experiments have actually been performed and they support SPIN and TWIN.

Conway stated that although he believed FIN to be true, he pointed out that, experimentally, FIN is the most contentious of the three axioms. It cannot be verified experimentally. The theories of relativity state that the speed of light c is the upper bound on the speed at which information transfer occurs. FIN does not require the theory of relativity to be correct (FIN requires any upper bound not necessarily c) although it would be sufficient. "We do not know if some unknown method allows for instantaneous transfer of information", Conway laughed, "almost by definition."

Kochen-Specker Paradox

Conway then went on to describe a simple version of the Kochen-Specker paradox. This paradox is a consequence of the SPIN axiom.

A particle being
measured from three orthogonal directions

When measuring the spin of a spin-1 particle along a direction, the values that are possible are:

parallel to the direction (+1)
perpendicular to the direction (0)
anti-parallel to the direction (-1)

As stated earlier, the SPIN axiom talks about the square of the spin and thus the values are limited to 1, 0 and 1.

Suppose a particle has already decided its spin in every direction. When an experimenter measures its spin in some direction the particle simply "answers" with the value of spin it has predetermined. Conway showed that this is not possible because there is no way to assign 0s and 1s to all directions that we can measure a particle from and still be consistent with the SPIN axiom. In his talk, Conway showed that even if an experimenter was limited to merely 33 directions, there was no way for a particle to predetermine the square of its spins in all 33 directions and still be consistent with SPIN.

Imagine a cube that snuggly surrounds a sphere. On each face of the cube, we inscribe a circle and inside each circle we draw a square that touches the circle at the squares four corners. We divide each such square into four smaller squares and mark the following points on the cube.

We get 33 points (9 points per face x 3 faces + 1 point per edge x 6 edges) on the cube in this way. These represent 33 directions of measuring a particle.

Lets try to assign a possible set of values of 0 and 1 to these 33 points consistent with SPIN. First note that we only need 33 points and not 66 because whether you measure a particle from one direction or from exactly the opposite direction, we get the same value. This is because we are measuring not spin, but spin squared (so 1² = -1² and 0² = 0²).


Without loss of generality let us assume we get a measurement of 0 when looking towards the particle from the center of one of the cube faces. Then by the 101 principle we know that the measurement from the direction of the remaining faces must be 1 because those faces are orthogonal.		In fact every point on the plane through the middle of the cube shown is orthogonal to our initial direction and hence must give a squared spin of 1.

If we continue in this fashion we deduce the spin that would be measured on a particle if it is to satisfy SPIN. However, having successfully deduced the spin for 32 of the directions (marked either green for 0 or red for 1 in the above diagrams), we find these 32 measurements and SPIN force the final measurement (marked yellow) of spin to be to be both 0 and 1 which is impossible!

There is no way to assign a spin value to each of the 33 directions that we have chosen to measure this particle in a manner consistent with SPIN. There is a nice Python script that allows you to interactively try out this experiment available here. This script was used to generate the pictures above.

What does this mean? It means one of two things:

Each measurement of a particle is not independent but rather depended on context. In other words, the order in which you make measurements matters and the value of a particles spin in a given direction depends on the history of measurements of that particle in other directions. The measurements are not commutable.
Alternatively, the particle does not decide what the value of its spin is in any direction until the experimenter actually makes a measurement!

This result is known as Kochen-Specker paradox and was discovered in 1967 by Simon Kochen and Ernst Specker. Conway said that while this was an interesting result, the hypothesis that measurements are commutable is untestable.

"Once you step into a river, you cannot step into that river a second time because in some sense its now a different river."

EPR Paradox

Next Conway talked about the EPR Paradox and the TWIN axiom. In 1935, Albert Einstein, Boris Podolsky and Nathan Rosen proposed a thought experiment hoping to demonstrate what they felt was a short-coming of quantum mechanics. This thought experiment is called the EPR-paradox.

Imagine two particles that are allowed to interact initially. We measure their total angular momentum or spin we will get a value between -2 and 2. Once in a while when we are lucky we get a total angular momentum of 0. This means we have a pair of particles whose individual spins sum to zero — we may have a particle with a spin of 1 and another with -1, or we may have two particles with 0 spin. Notice if we square the particle spins, we get identical values for each pair. What this says is that if two particles total angular momentum is zero, then the square of the spin of one particle measured from some direction is identical to the square of the spin of the other, irrespective of how far apart these particles may drift before they are measured.

The paradox that Einstein and his colleagues desired to demonstrate was that making a measurement in one part of a quantum system can have an instantaneous effect on measurements made elsewhere in the system.

Conway-Kochen Proof

Finally Dr Conway had set up sufficient background to deliver his proof.

Suppose we have two particles P_home and P_away. Suppose further that these particles together have a zero angular momentum. We allow P_away to drift away from P_home so that the two particles become separated by some large distance. Now we would like to measure P_home from some three orthogonal directions namely x, y and z. Also we would like to measure P_away from a direction ω where ω is the same as one of x, y or z.

Lets measurement of the particle P_home and P_away be represented as two functions f_home and f_away. The output of this function may depend on:

the particular direction we are measuring
the order in which we make our measurements
additional information available only to P_home
additional information available only to P_away
additional information available to both P_home and P_away.

Note the aspect that makes this proof novel is that we allow the particles any amount of additional information (within the constrains of our three initial assumptions) including their past histories and ability to communicate. Conway said "We can even imagine that the particles could be 'listening' to us the experimenters as we decide the order, they could be communicating in some way or whatever." We will show that this information cannot help the particles "decide" ahead of time what values to return. Instead the particles spin values cannot be decided until the experimenters have decided what directions they will measure the particles from.

Consider the two functions:

f_home(y, (z, x, y), I_A, I_B, I_AB)
f_away(ω, ( ω ), I_A, I_B, I_AB)
where
- the first argument is the direction that is being measured
- the second argument is the order in which the directions will be measured
- the third argument is information available only to P_home
- the fourth argument is information available only to P_away
- the final argument is information available to both P_home and P_away
By the FIN axiom, information transfer between the two particles occurs at a finite speed. Here is a spacetime diagram showing the two particles and the information that is accessible to each in time.

Once the particles are separated by distance , it takes a time T for information from one particle to reach another. The information shown as I_A in the diagram is available only to P_home and not to P_away. If we do our experiment within time T then P_away has no access to I_A so I_A cannot possibly affect the output of f_away. Similarly, P_home cannot access I_B so I_B cannot affect the output of f_home. So we can remove these variables from our measurement function:

f_home(y, (z, x, y), I_A, ~~I_B~~, I_AB)
f_away(ω, ( ω ), ~~I_A~~, I_B, I_AB)

We now choose to measure the two particles from the same direction. Lets choose y = ω.

f_home(y, (z, x, y), I_A, ~~I_B~~, I_AB)
f_away(y, ( y ), ~~I_A~~, I_B, I_AB)

Every time we do this experiment the information available to both particles, I_AB is different. But whatever that information is, for a given experiment, it is fixed and identical. So for a given measurement f_home₀ and f_away₀, let the value of I_AB be a constant i₀.

f_home₀(y, (z, x, y), I_A, ~~I_B~~, i₀)
f_away₀(y, ( y ), ~~I_A~~, I_B, i₀)

By the TWIN axiom, we know that if we measure the two particles from the same direction, the result will be the same:

f_home₀(y, (z, x, y), I_A, ~~I_B~~, i₀) = f_away₀(y, ( y ), ~~I_A~~, I_B, i₀)

Now Conway put forth an argument to show that f cannot depend on I_A or I_B either. The argument is straightforward but can be difficult to follow if you unfamiliar with it. f_home₀ does not depend on I_B so varying I_B makes no difference to the value of f_home₀ — its constant with respect to I_B. But the functions are equal, so f_away₀ is also constant with respect to I_B. In other words, changing I_B must have no effect on f_away₀ and we can remove it as a parameter from f_away₀.

f_away₀(y, ( y ), ~~I_A~~, ~~I_B~~, i₀)

The same argument holds for I_A and f_home₀.

f_home₀(y, (z, x, y), ~~I_A~~, ~~I_B~~, i₀)

In other words, the spin of a particle is dependent solely on the direction from which it was measured and not on its history. But we have already seen from the Kochen-Specker paradox, there is no way for a particle to predetermine its spin in every direction in a way consistent with SPIN.

Conway thus concluded that if the experimented had sufficient freewill to decide the directions in which he would measure the particle then the particle too must have the freewill to decide on the value of its spin in those directions such that it can be consistent with the 101-property. In concluding Dr Conway said that he believed he did have freewill. Holding up a piece of chalk, he said he felt he could choose whether or not he would drop it or continue to hold it. His theorem he said leads him to accept that the universe is teeming with freewill. He also said that while he did not have any proof for it, he believed that the cumulative freewill of particles is the source of his freewill as a person.

Questions

When the floor was opened for questions, one member of the audience questioned Dr Conway's use of the term "Free Will". She asked whether Dr Conway was "confusing randomness and free will".

In a passionate reply, Dr Conway said that what he had shown, with mathematical precision, that if a given property was exhibited by an experimenter than that same property was exhibited by particles. He had been careful when constructing his theorem to use the same term "free will" in the antecedent and consequent of his theorem. He said he did not really care what people chose to call it. Some people choose to call it "free will" only when there is some judgment involved. He said he felt that "free will" was freer if it was unhampered by judgment - that it was almost a whim. "If you don't like the term Free Will, call it Free Whim - this is the Free Whim Theorem".

I would like to thank Morris W. Hirsch for correcting a typographical and presentation error in this document.

Thank you to Frank Foreman for sending me the following picture.
Yield to Whim