We consider experimentersA andB performing the pair of experiments described in the TWIN axiom on separated twinned particles a andb, and assert that responses ofaand b cannot be functions of all the information available to them.
Let’s first show that the value θ(w) of “the squared spin in direction w” doesn’t (al-ready) exist prior to its measurement, for if it did, the function θ(w) would be defined for each direction w, and would have according to SPIN the following interesting prop-erties:
1. Its values on each orthogonal triple would be 1, 0, 1 in some order.
This easily entails two further properties:
2. We cannot have θ(x) =θ(y) = 0 for any two perpendicular directions x and y.
3. Because θ(w) is “the value of the SQUARED spin in direction w”, we haveθ(w) = θ(−w) for any pair of opposite directions w and −w.
Consequently, θ(w) is really defined on “± directions.”
We call a function on a set of directions that has all three of these properties a “101-function.” However, we already know from the Kochen-Specker paradox (proven above for Peres’ 33-direction configuration) that this certain geometric combinatorial puzzle has no solution. This disposes of the above naive supposition about existence of θ(w) prior to its measurement.
We are now left only with the possibility that the particles’ responses are not prede-termined (ahead of time), but we will try to prove that they also can not be a function of the information available to them.
So let us suppose, that particle a’s response is a functionθa(α) of the information α available to it and show that this assumption leads to contradiction. Information α is determined by the choice of the certain triplex,y,z and all the information α0 that was accessible just before and independent of that choice, i.e. the information in the past
light cone of a (FIN).4 So we can express it as a function θa(x, y, z;α0) ={x→j, y→k, z →l}
or if we refine this notation to indicate the result of measurement in any particular one of the three directions by adjoining a question–mark to it, thus
θa(x?, y, z;α0) = j θa(x, y?, z;α0) = k θa(x, y, z?;α0) = l
Similarly we express b’s responses as a function of the direction w and the information β0 available tob beforew was chosen
θb(w;β0) ={w→m}
or alternatively
θb(w?;β0) = m The TWIN axiom then implies that
θb(w?;β0) =
θa(x?, y, z;α0) if w=x, θa(x, y?, z;α0) if w=y, θa(x, y, z?;α0) if w=z,
According to the Free Will AssumptionA and B can freely choose any direction w and triple of orthogonal directions x, y, z from Peres’ set of ±33 directions, but whenever the directions coincide, the corresponding measurements must yield the same answers regardless of the informationα0 and β0. Now, the important thing is, that not only is the informationα0by definition independent of the choice ofx,y,z, but it is also independent of the choice of w, since in some coordinate frames B’s experiment happens later than A’s and in order to preserve causality can not influence A. For the same reason, β0 is independent of x, y, z as well asw.
Now, to conclude the proof, let us first notice that responses of paricles a and b to measurement of the squares of components of spin are of the form
θa=θa(x, y, z;α0)6=f(w;β0) θb =θb(w;β0)6=f(x, y, z;α0)
but at the same time have to fulfil TWIN, therefor it is evident that they can only be fuction of direction of measurement, independently of the measurement arrangement, i.e.
4even if some additional piece of information is created after the choice of the triple, it must clearly be of formi(x, y, z;α0), meaning that we can still describe response ofaas some function ofx,y,zand α0: θa(x, y, z;α0, i(x, y, z;α0)) =θa0(x, y, z;α0)
the response is noncontextual. To see that, let us consider for example experimenter A measuring the triple x,y,z and obtaining in xdirection the result
θa(x?, y, z;α0) =j
From TWIN it follows, that if experimenter B also measured in the same direction, he would obtain the same answer
θb(x?;β0) = j
and because that response is not a function of x,y,z, he would obtain it regardlessly of that, what kind of measurement (x, y, z) if any at all performed A. But on the other hand this entails that also A’s measurement of any other triple that includes x would produce the outcome j in xdirection
θa(x?,ey,ez;α0) =j
because it must correspond to possible, but at the same time independent B’s mea-suremnt. We have thus proven that in every single case (for any fixed α0, β0) the out-comeθ(w) is completely determined for any direction.w5 Because of the SPIN it must be described by the “101-function” of direction, which doesn’t exist for Peres’ configuration.
So we arrived at contradiction. The crucial moment in this derivation was to realize, that the outcomes on both sides are independent of each other but at the same time because of quantum entanglement yield same values in the same directions, so our inital assumption that responses of particles are a function of our choice of a triple turned wrong.
We have thus first refuted the possibility that the outcomes of measurement are predetermined, and now also the case in which they are determined by its history and our free choice (triple x, y, z and the state of the universe before the choice of that triple). This leaves only the case in which some of the information used (say, by a) is spontaneous, i.e. is itself not determined by any earlier information whatsoever. This spontaneous information arises in the moment when the universe takes a free decision and contibutes an additional input to the result of the measurement.
This completes the proof of the Free Will Theorem, except for a brief remark on that in the statements of axioms used above and throughout the proof we have made some tacit idealizations that might worry someone. For example, we have assumed that the spin experiments can be performed instantaneously, and in exact directions. Both assumptions and subsequent proof can be replaced by more realistic ones that account for both, the approximate nature of actual experiments and their finite duration. The idea is to take into account that the twinned pair might only be nominally in the singlet state and that orthogonality of frame (x, y, z) and parallelism betweenw and one of the triple x, y, z might be only approximate. Redefined SPIN and TWIN axioms take into consideration that expected ideal results are achieved only in a portion of experiments.
By estimating inaccuracy in angle it is possible to show, that with sufficient precision, the probabilty of discrepancy of outcomes from those obtained in the ideal conditions can be dropped below 1/40. Namely any function of direction must fail to have the 101–property
5θa(w) is completely determined by possible measurementθb(w;β0)
for at least one of 40 particular orthogonal triples (the 16 orthogonal triples of the Peres’
configuration and the triples completed from its remaining 24 orthogonal pairs).
So, what we have proven is, that if there are experimenters with a modicum of free will, then response of elementary particles is also free. Close inspection reveals that Free Will Assumption was only needed to force the functions θa and θb to be defined for all of the triplesx, y, z and vectorsw from a certain finite collection and some fixed values α0 and β0 of other information about the world. Now we can take these as the given arbitrary initial conditions that enter into some putative free state theory that explains the results of the measurement and then the same procedure proves also the Free State Theorem.
Chapter 5
Conclusion
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Chapter 6 Appendix
The Proof of commutation of squares of components of spin in three orthogonal directions for S=1 particles
In case of S=1/2 or S=1: [ˆS2i,ˆS2j] = 0 ; ∀i.j ∈ {x, y, z}, but in general this doesn’t hold true.
To show that, we take into consideration the following rules for commutators:
[ˆSi,ˆSj] = i~εijkSˆk hSˆ2,ˆSii
= 0 hSˆz,ˆS±
i
= ±~Sˆ±
hSˆ±,ˆS∓
i
= ±2~Sˆz hˆS2,ˆS±
i
= 0
...and those for operators:
ˆSz|s, mi = ~m|s, mi
Sˆ2|s, mi = ~2s(s+ 1)|s, mi Sˆ±|s, mi = ~p
s(s+ 1)−m(m±1)|s, m±1i; Sˆ± =ˆSx±iSˆy
[AB, C] =A[B, C] + [A, C]B =⇒[A, BC] =B[A, C] + [A, B]C S = 1 =⇒(2S+ 1 = 3 : M =−1,0,1)
Sˆ2z|1, Mi=~2M2|1, Mi ∈ {0, ~2|1, Mi}
Sˆ2|1, Mi= (Sˆ2x+Sˆ2y+Sˆ2z)|1, Mi=~2S(S+ 1)|1, Mi= 2~2|1, Mi
So, we see that when measuring the squares of the components of spin in three orthogonal directions, they yield the outcomes 1,0,1 in some order.
ˆS+|1,−1i = √
We use the matrix representation of operators A|Ψiˆ =P
For S=1 this expression becomes: