Slide 1

The Conway-Kochen "Free Will Theorem"

Nicholas Wheeler
Reed College Physics Seminar
2 December 2009

Slide 2

Deep History

EINSTEIN, PODOLSKY & ROSEN (EPR)

1935: A. Einstein, B. Podolsky & N. Rosen "Can quantum-mechanical description of physical reality be considered complete?" Phys. Rev. 47, 777 (1935).

     

Written primarily by Podolsky. Rosen (then 23 years old), who had acquired prior experience with "entangled states"—yet to be given their name—from work on the hydrogen molecule.

Paper pointed out that if a composite system (comprised of subsystems A and B), is in an entangled state, then the results of measurements performed on A and B will exhibit correlations, even though the measurement events lie outside each other's lightcones. This held by Einstein, a "local realist," to be absurd—spoke of a "spukhafte Fernwirkung" (spooky action at a distance), rendered quantum mechanics implicitly nonlocal.

"I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming house, than a physicist". —AE

Slide 3

Problem avoided if particles were endowed with internal "instruction sets," written onto "hidden variables." Probabilistic aspects of quantum mechanics would then be a reflection of our ignorance of the state of the hidden variables…and it was that kind of ignorance that in EPR's view rendered quantum mechanics "incomplete."

Bohr immediatey wrote a strong rebuttal (which Bell later held to be "almost unintelligible"), and anyway it was widely believed that John von Neumann (1932) had shown hidden variable theories to be impossible (by a line of argument that Bell later showed to contain a fatal flaw).
  
So Pauli, Born and nearly everybody else was happy to dismiss the whole issue.  
  
Schrödinger, however, considered (1935) that EPR had identified the most profoundly puzzling/distinctive feature of quantum theory (and gave "entangled states" their name).

Slide 4

DAVID BOHM, YAKIR AHRANOV

1951: Publishes Quantum Mechanics, Part 3 treats foundational problems, emphasizes importance of EPR problem.

1957: David Bohm & Yakir Aharonov publish "Discussion of experimental proof for the paradox of Einstein, Rosen & Podolsky" in Phys. Rev.—reformulate the EPR argument in terms of two spin one-half particles in an entangled singlet state.

Experiments show "spooky action-at-a-distance to be real."

Slide 5

BELL'S THEOREM

1964 : J. S. Bell publishes "On the Einstein Podolsky Rosen paradox," Physics 1, 195 - 200.

Uses two spin one-half particles in singlet state (a la Bohm/Aharonov) and spin measurements in three non-colinear directions. Hidden variable analysis leads (as Bell ingeniously demonstrates) to an inequality which ordinary quantum theory would in some cases violate, the implication being that

"No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."

Bell had identified von Neumann's elementary error, made lively again the possibility that hidden variables might exist. Tended intuitively to favor that hypothesis, sought to establish that quantum mechanics is a local theory—incomplete as it stands. But…

Experimental work (1970s-80s), by Alain Aspect, Anton Zeilinger and others, established quite convincingly that (more workable generalizations of) Bell's inequality are in fact violated: non-locality, spooky action at a distance are facts to be lived with.

But experimentalists were kept busy for a decade or more attempting to close various subtle loopholes. Some of those have to do with the fact that the objects on both sides of Bell's inequality are statistical averages: any finite set of data might therefore be misleading.

Slide 6

GLEASON'S THEOREM

1957: George Mackey (1916-2006)—thesis advisor (1973) of Thomas Wieting—did pioneering work on the mathematical/logical foundations of quantum mechanics: his Mathematical Foundations of Quantum Mechanics was published in 1963.

Andrew Gleason (on the left) wrote his thesis under Mackey (1950), and in (1957) produced Gleason's theorem ("one of most basic/important 20th Century contributions to quantum mechanics"). Not directly concerned with hidden variable question, but sprang from the same von Neumann/Mackey quantum axiomatics that led von Neumann to his (fallacious) "no-hidden variables theorem."

Bell recognized the importance of Gleason's theorem, of which he provided his own alternative proof.

Slide 7

KOCHEN-SPECKER THEOREM

1967: Simon Kochen & Ernst Specker publish "The problem of hidden variables in quantum mechanics," Journal of Mathematics & Mechanics 17, 59-87. The work had been completed in 1963-64—simultaneously with but in ignorance of Bell's work.

The Kochen-Sprecker theorem is the first of what has become a large population of "no-go theorems"—theorems that demonstrate in a variety of contexts that the conditions imposed on the "hidden variable instruction sets" pose combinatorial problems that possess no solution. 
   
The KS theorem excited physicists (tiny population of "foundations physicists") because it provides an instance of a non-statistical state-independent proof of the impossibility of "non-contextual" hidden variable theories.
   
The KS theorem can be looked upon as a simplified derivation of an implication of Gleason' s theorem which, however, was unknown to Specker when he first took up work in this area (1960).

The KS theorem—the work of two "applied logicians"—involves an intricate argument concerning the orthogonality relations present in a certain collection of 117 rays in 3-space, and does not look much like "physics"!  Attempts were made to simplify the line of argument that KS had introduced—initially by F. J. Bellinfante in his A Survey of Hidden Variable Theories (1973). 

So far as is known, the simplest 3-dimensional version of the KS argument (Conway & Kochen: 2002) employs a collection of 31 rays. Much simpler—but weaker—versions (by David Mermin and others) live in Hilbert spaces of higher dimension. 

For a sense of the density of publication in this area (particularly by the Chinese!) see

Hyperlink["QM Foundations Bibliography","http://www.scribd.com/doc/19295119/Bibliographic-Guide-to-Foundations-of-Quantum-Mechanics-A-Cabello" ]

which—fairly complete up to May 2005—cites 10,340 papers! Foundational physics has become a crowded field!!

The simplifications that result from appeal to the quantum physics of spin-one particles (see below!) were first noted in unpublished work (1969) by Richard Friedberg at Columbia University, according to Max Jammer,  The Philosophy of Quantum Mechanics (1974), page 324.

Slide 8

ASHER PERES' SIMPLIFIED VERSION OF THE KOCHEN-SPECKER NO-GO THEOREM

1991: Asher Peres (1934-2005)—a student of Nathan Rosen

—publishes an attractively symmetric version of the KS argument that involves only 33 rays in 3-space. The rays in question are generated by the following figure:

I return to Peres' argument after a brief…

Slide 9

DIGRESSION: ROGER PENROSE ENTERS THE PICTURE

Roger Penrose (~1992) heard Peres' describe this work at a conference in Copenhagen, asked Peres about the geometrical significance of his set of rays. Peres responded that he had not thought about it. Penrose quickly realized that they are the symmetry axes of this figure

Slide 10

This figure was for Penrose was an old friend: it appears in Maurits C. Escher's "The Waterfall"

in which Escher developed an idea suggested to him years before by Penrose's father! The story is told on page 30 of the February 1993 issue of Scientific American.

Penrose was inspired by Peres' accomplishment (and enabled by his familiarity with Ettorre Majorana's little-known spin formalism) to develop his own elegant no-go theorem:

Slide 11

Hyperlink["Majorana Biography","http://en.wikipedia.org/wiki/Ettore_Majorana"]

J. Zimba & R. Penrose, "On Bell non-locality without probabilities: more curious geometry," Stud. Hist. Phil. Sci. 24, 697, (1993). See also J. E. Massad & P. K. Aravind, "The Penrose dodecahedron revisited," AJP 67, 631 (1993). It was a Tom Wieting seminar based upon the latter paper that first brought Penrose's work to my attention.

Penrose's theorem exploits properties of 40 rays that live in a certain
 4-dimensional Hilbert space.  20 of those—which Penrose calls "ordinary rays," refer to the spin-one-half states that result when the spin of a spin-3/2 particle is measured along one or another of the 20 directions defined by the vertices of a dodecahedron. Here again, cubes play a central role in the discussion: the dodecahedral vertices are organized into subsets by the five cubes that can be inscribed within a dodecahedron, and by the two tetrahedra that can be inscribed within each such cube:

To his "ordinary rays" Penrose adjoints the 20 "extraordinary rays"—one associated with each vertex—defined by their orthogonality to the ordinary rays at the three nearest neighbors of that vertex.  From these ingredients, Penrose fashions what he calls a "Quintessential Toy," in terms of which he poses an insoluable coloring problem. This he uses in Chapter 5 of Shadows of the Mind (1994) to dramatize the fact that "quantum mechanics in in some fundamental respects very strange, very counterintuitive." 

Ultimately, Penrose places two such spin 3/2 particles in a singlet state, gives one to Alice, one to Bob, constructs an argument very like Bell's before him, and very like the argument adopted later by Conway & Kochen.

End of digression.

Slide 12

PERES' PROOF OF THE KOCHEN-SPECKER THEOREM

…exploits a remarkable property of spin-one particles:

The squared spin matrices refer to a compatable triple of measurements:

THE 101 RULE: For spin-one particles, spin-squared measurements in three orthogonal directions always produce some permutation of {1, 0, 1} (in units where ℏ = 1).

• For spin one-half this reduces to a triviality:
(1/4)+(1/4)+ (1/4)= (3/4) 

• For spin ⩾ (3/2) such a result is impossible, because the spin-squared operators fail to commute.

So we will (with Peres) be exploiting a fact that pertains uniquely to (massy) spin-one particles.

Slide 13

You are a local realist, seek to write non-contextual instructions onto variables hidden within the spin-one particle that say how the particle is to respond—consistently with the 101 rule—to every possible triadic array of spin-squared meters (of which there are infinitely many).

Here is a simple solution in 2D

That simple idea fails in 3D

but to establish that failure is inevitable requires a more subtle line of argument.

Peres uses a population of only 33 rays—16 orthogonal triads—to establish that such an assignment cannot be accomplished. From the following population of 33 rays

one can assemble 1 + 6 + 6 + 3 = 16 orthogonal triads:

Slide 14

Some rays are members of a single triad, some of two, some of four. The graph shows which are members of which:

The ● dots refer to center-of-face rays, the ● dots to center-of-edge rays. All triads are represented by triangles (and vice versa)  except for the three ●-●-● triads.

The objective of Peres' game is to paint red one ray in every triad, subject to the constraint that (not just its companions in the triads of which it is a member, but) every ray orthogonal to a red ray is painted black. Orthogonal rays cannot be both red.

This looks like a solution:

Slide 15

Here is a list of the 72 orthogonal pairs that can be constructed from Peres' set of 33 rays:

And here is a list of the rays we have proposed to paint red:

That creates

in which we encounter {4 ■, 27 ■} and {11 ■, 24 ■}—orthogonal pairs of which both members are red, in violation of the rules. So it goes in every case.

UPSHOT OF THE KOCHEN-SPECKER-PERES THEOREM: It is impossible/inconsistent to require of a function

that it be a "101 function"—a function that yields

for all orthogonal triadic arguments: no such function exists.

The Kochen-Specker-Peres proof that non-contextual hidden variables (for the system at hand, and therefore universally?) cannot exist is non-statistical, and makes no assumption regarding the initial state of the spin-one particle, and in both respects represents an improvement upon Bell's proof of the same result.

The KS theorem does not exclude the possibility of "contextual hidden variables." This is a loop-hole that Conway & Kochen close en route to their "Free Will Theorem."

Slide 16

The Free Will Theorem

July 2006: John Conway & Simon Kochen (Princeton Mathematics Department) publish 

"The Free Will Theorem," Foundations of Physics 36, 1441–1473 (October 2006)

"If the choice of directions in which to perform spin-one experiments is "free"—not a function of information written into the backward lightcone of the experimenter—then the responses of the particles are similarly free."

Conway & Kochen were motivated by criticism of their initial paper to modify their argument: "Reply to comments of Bassi, Ghirardi, and Tumulka on the free will theorem: The Strong Free Will Theorem," Notices of the AMS 56, 226–232 (February 2009).

Conway & Kochen exploit a remarkable property of paired spin-one particles in …

THE SINGLET STATE

Notice that 0 is a shared eigenvalue (and the only non-degenerage eigenvalue of SS).

The associated eigenvector is

The associated density matrix is

If we introduce the eigenvectors of (Subscript[s, 3])

we find that the singlet state can be described

so is a manifestly entangled state.

Slide 17

Spin-one particles in the singlet state are "TWINNED"

Particles A and B are in the singlet state. Particle A is given to Alice, particle B is given to Bob. 

The meter with which Alice measures "spin squared" in direction a is described by the matrix

Similarly, to describe Bob' s meter ("spin squared" in direction b) we write

Alice's spin-squared meter announces "0" with probability

and leaves the composite system in the state

Bob's meter now announces "0" with probability

and announces "1" with probability

The implication is that if Alice gets "0" and if Bob's meter is precisely aligned with Alice's, then Bob also gets "0" with certainty. Moreover, if Alice has a triadic set of meters, and they announce "1, 0, 1" in some order, and if Bob's tradic set is identically oriented, then Bob will get "1, 0, 1" in the same order.

These guaranteed coincidences presume the initial state to be the singlet state, and are what Conway & Kochen mean when they say the particles are "TWINNED".

Slide 18

FAILURE OF THE CONTEXTUAL HIDDEN VARIABLE HYPOTHESIS

We propose to escape from the force of the KS theorem by assuming that the hidden variables (instruction sets) refer to orthogonal triads, rather than to individual rays. Spin-squared measurement in a given direction a will therefore be 0 or 1 depending upon the triadic context of the a-measurement.

Conway & Kochen provide Alice with one of a twinned pair of spin-one particles, and Bob with the other particle.

They equip Alice with a 16-button device, each button serving to align her triadic arrangement of spin-squared meters with one or another of Peres' 16-member set of orthogonal triads. 

Bob is similarly equipped, but each of his 33-buttons has sub-buttons that indicate which of the indicated triad of meters will be read, which ignored. In effect, Bob has a 33-button device, each button serving to align a solitary spin-squared meter with one or another of Peres' 33 ordinary rays.

Conway & Kochen assume that Alice & Bob, their respective particles and laboratories lie far outside of each other's lightcones; i.e., are separated by an arbitrarily large space-like interval:

Note that it makes no sense in a relativistic world to ask "Who made the first measurement?" Conway & Kochen take for granted what they call the "causality principle": effects cannot—in the opinion of any inertial observer—precede their causes.

Alice elects "of her own free will"—meaning in a way not predetermined of dictated by any of the events in her backward lightcone—to press the (i^th) button, which aligns her meters along {(Subscript[a, i]),(Subscript[b, i]),(Subscript[c, i])}.  Her meters read

Bob elects of his own free will to measure spin-squared in direction (Subscript[x, k]): k=1,2,…,33. His selection cannot have been influenced by Alice's, nor can the response (0 else 1) of his meter have been influenced by the responses of Alice's meters.

By the Kochen-Specker theorem we know that there exists at least one direction  x which is an element of two distinct triads  {x,a,b} and  {x,c,d}, and that it is instructed by the conjectured contextual hidden variables to register 0 when a member of one triad, 1 when a member of the other. But if Bob selects  x  his meter will read

So he is at risk of violating the twinning principle.

Phrased another way, we must, by twinning, have

But according to the Kochen-Specker theorem there is no function (Subscript[θ, 0])[x] that assigns compatable 0-1 values to simultaneously to all 33 Peres rays.

Slide 19

FAILURE OF DETERMINISM—ENFORCED FREE WILL?

The Kochen-Specker theorem was seen to render the non-contextual hidden variable hypothesis untenable. And it has been shown by Conway/Kochen to render the contextual hidden variable hypothesis equally untenable (at least for twinned spin-one particles, therefore generally?).  Conway/Kochen argue that those two hypotheses exhaust the possibilities, that no mechanism for exercising hidden deterministic control of quantum processes survives. The quantum world has been shown to be indeterministic.

Therefore "quantum systems do not know how they will respond to measurement until they have done so." Which echos Bohr's "No quantum phenomenon is a phenomenon until it is a recorded phenomenon."

Alice/Bob did not know which buttons they were going to press until—unforcedly/undeterministically—they did so. Which is what we mean when we say they acted from spontaneous exercise of free will. Conway & Kochen insist that for that same reason, and to that same extent, we must attribute "free will" to the particles themselves. Which is the upshot of their FREE WILL THEOREM.

Slide 20

COMMENTARY

Conway & Kochen do not suppose particles to be sentient—they have detached "free will" from "mind," and therefore from all that philosophers have had, over the centuries, to say about "the free will problem."

Is the traditionally troublesome concept of "free will" so well-understood that its use by physicists can serve to clarify understanding of (say) the quantum mechanical two-slit experiment? Or any quantum phenomenon?

Hyperlink["What Philosophers Say about Free Will","http://plato.stanford.edu/entries/freewill/"]

Hyperlink["A Philosopher on Conway-Kochen","http://philosophy.ucsd.edu/faculty/wuthrich/PhilPhys/MenonTarun2009Man_FreeWillThm.pdf"]

Philosophers (Hobbs, Hume, Poincaré, …) have been at pains to separate "free will" from "determinism," to carve a place for free will in a deterministic will. Whence the several flavors of "compatibilism":

Hyperlink["Compatibilism","http://plato.stanford.edu/entries/compatibilism/"]

Such philosophers, having established that conceptual independence, would insist that indeterminism does not imply free will. One cannot, in particular, identify free will as a necessarily active agent in a random world.

Conway & Kochen tacitly dismiss the possibility that Alice/Bob and their particles/equipment might be interlinked by tachyons. No problem if tachyons cannot convey information…as has been argued by those to assert the impossibility of "tachyonic telephones" (which would enable one to communicate/interact with one's past).

Hyperlink["Tachyonic Telephones","http://en.wikipedia.org/wiki/Tachyonic_antitelephone"]

One can imagine a world so deterministic that Alice/Bob are prevented from making the choices that would expose results inconsistent with determinism. Conway & Kochen dismiss this possibility on grounds that in such a conspiratorial world (where the free decisions of experimentalists are constrained) science is impossible.

Since about 1990, Gerard 't Hooft has been cultivating a theory which purports to deduce quantum theory from a completely deterministic physics at the Planck scale. He claims that the Conway/Kochen argument fails because Alice/Bob actually do NOT have free will they imagine themselves to have…so therefore neither do particles.

Hyperlink["Determinism Beneath Quantum Mechanics","http://arxiv.org/pdf/quant-ph/0212095v1"]

Hyperlink["'t Hooft's Opinion","http://www.sciencenews.org/view/generic/id/35391/title/Math_Trek__Do_subatomic_particles_have_free_will%3F"]

Physicists sometimes allow themselves to say "the system wants to …",  "Nature might have been that way, or that other way, but chose to be this way…".  I try never to use, and always object to such anthropocentric language, but not so Conway (& Kochen), who carry it to a shameless extreme. It is my own suspicion that the laws of Nature cannot self-consistently be other than what they are, and that—contrary to the claims of some string theorists—no question of "choice" is involved. Interestingly, Conway and other mathematicians never assert that revealed mathematical facts "might" have been other than what they are found to be, and become nervous when mathematical arguments appeal to the Axiom of Choice.

Conway and Kochen can properly claim to have driven yet another nail into the hidden variables coffin, though some people find loopholes in their argument, to which they take technical exception:

Hyperlink["Tumulka's Critique","http://arxiv.org/pdf/quant-ph/0611283"]

Hyperlink["What Does the Free Will Theorem Actually Prove?","http://arxiv.org/abs/0905.4641"]

Conway & Kuchen take their "discovery" very seriously, stress that it is after all a "theorem". Are reportedly writing a book-length account of the subject, to be published by Princeton UP. 


But in my view all they have done is to decorate an old cake with some (objectionably anthropomorphic) rhetorical frosting. Conway's work has been marked by breathtaking originality, a calculated capacity to astound. But here he seems to have fallen victim to his own sophomoric enthusiasm. His mathematical innovations have taught us useful things we didn't already know. But his Free Will Theorem, though it is based upon sound/interesting physics, seems NOT to teach us "useful things we didn't already know," and to have no useful consequence.


My guess is that the Conway/Kochen Free Will Theorem is destined to be kicked around by philosophers/kooks…and ignored by physicists—except, perhaps, in this regard:

Experimentalists who based their work on Bohm/Ahranov used paired spin-one-half particles in a specialized state (the singlet state) to extablish violation of Bell's inequality—evidence that non-contextual hidden variables cannot exist. Kochen-Specker made no state-specializing assumption, and very recently (May 2009) experimentalists have provided state-independent verification

Hyperlink["State-independent Experimental Non-contextuality","http://arxiv.org/abs/0904.1655"]

of the correctness of the Kochen-Specker conclusion that non-contextual hidden variables cannot account for the quantum mechanical facts. Perhaps a state-independent experimental dismissal of contextual hidden varables lies not too far in the future.

The Conway-Kochen paper can be found at

Hyperlink["Free Will Theorem","http://arxiv.org/abs/quant-ph/0604079" ]

For videos of six lectures on the subject, presented before non-specialist audiences at Princeton (23 March––27 April 2009) by Conway himself, visit

Hyperlink["Conway Lectures", "http://www.princeton.edu/main/news/archive/S23/75/58A30/index.xml?section=featured"]

and click on viewing online.

Hyperlink["Google","http://www.google.com/"]

Key word: free will ("free willy" gives 3,910,000 hits, "free will astrology" gives 9,590,000 )

Key word: free will theorem (10,300 hits, ~10,000 kooky blogs)