## The Kochen-Specker Theorem

I will now take a break from the prior series, and discuss a few fundamental results in quantum mechanics which were not given proper attention in the past on this blog. Today I am discussing the Kochen-Specker theorem, which rivals in importance Bell’s theorem.

When you read about this result the first time, it looks a bit dry and abstract, but in fact it is child’s play because it is nothing more than a coloring game. Originally the first proof was quite intricate, but later on, the late Asher Peres found a great simplification and I will discus this instead.

Before starting, we need one preliminary result: for particles of total spin 1, we can measure the square of the component of spin in a direction and get +1 or 0. So far nothing special, but quantum mechanics shows that if we perform such a measurement on three orthogonal directions (say on x, y, z) we will get two results of +1 and one result of 0. We do not know what result will be on which direction, but we will always get a zero result and two +1 results in some order. I’ll not prove this, but I want to look at its meaning instead.

We know that in quantum mechanics the results of experiments do not exist before measurement, but can we create a model which will recover the 1,1,0 prediction for spin one particles? This will work for three orthogonal directions, but what if are adding additional orthogonal directions? If particles do have definite properties before measurement, it should be possible in principle to pre-assign the values of +1 and 0 to all our measurement directions in such a way that the +1,+1, 0 theorem is obeyed. The Kochen-Specker theorem shows that this is impossible.

(I am adapting this from a famous paper about Free Will: http://arxiv.org/pdf/quant-ph/0604079v1.pdf )
So here is the explanation: start with a cube, inscribe a circle on each face and add a point on the 4 places it touches the face sides (e.g. points V,D, U, etc). Then add a point in the middle of each face (points X, Y, Z) and connect this with the vertexes of the square. Add a point where this line intersects the circle (C, B, C’, B’) and unite them in a smaller square. Draw perpendiculars from the center of the face to the small square and add 4 more points (e.g. D, D’). Finally connect the center of the cube with all those points and obtain 33 directions: 13*3 – 4*3/2 = 33 directions [13 point on a face*3 faces but all 4 points on the inscribed circle are counted twice]

Now we can start the coloring game and prove the Kochen-Specker theorem. We will color the 1’s as red, and the 0’s as blue and see if this can be done in general or not.

Step 1: X, Y, Z for 3 orthogonal directions:

(we can pick X as zero without loss of generality)

Step 2: X-A implies A = +1 because the center of the cube and X and A form orthogonal directions, and on any 3 orthogonal directions there can be only one zero which is at X now) In turn A’ = 1 because X, A, A’ form 3 orthogonal directions.

Step 3: A,B,C form 3 orthogonal directions (this shows the cleverness of Peres’ choice for directions, try to prove using simple geometry that A,B,C form 3 orthogonal directions). Without loss of generality we can pick B=1, C=0.

and from A’ B’ C’: B’ = 1 and A’ = 1

Step 4: orthogonality of CD implies D = 1. Similarly C’D’ implies D’ = 1

Step 5: Z, D, E orthogonality implies E = 0. Z, D’, E’ implies E’ = 0

Step 6: EF and EG orthogonality implies G=G = 1. E’F’ and E’ G’ implies F’ = G’ = 1

Step 7: F, F’, U implies U = 0

Step 8: G, G’, V implies V = 0

Now for the contradiction: U is orthogonal with V and you cannot have both of them equal with zero.

So what does this mean? This shows that we cannot have a context independent assignment of measurement outcome before measurement. If we want to pre-assign measurement properties, this can only be done within the context of the measurement setting. K-S theorem weakens the idea of objective reality independent of measurement.

1. "K-S theorem weakens the idea of objective reality independent of measurement."

No it doesn't. It weakens the idea that possibilies exist before actualisation. Measurement actualises possibilities. Some measurements are mutually exclusive. Therefore not everything that is possible can be actualized. However, this says nothing about objective reality independent of measurement. All it says is that mutually exckusive context dependent observables (possibilities) can not be simultaneously actual.

2. To me context independence is a cornerstone of objective reality. There are contextual models of quantum mechanical predictions but I don't find them satisfactory. But this is my preference and other people can have different opinions.

This post was only meant as a introduction into this topic an not as a exhaustive discussion.

For a complete discussion on the meaning of K-S theorem, I recommend http://plato.stanford.edu/entries/kochen-specker/

3. "To me context independence is a cornerstone of objective reality."

Yes, objective reality should not depend on measurement contexts. But you ate confusing observables with beables. All I'm saying is that context independent objective beables are completely consistent with contextual observables.

4. Let me first show that my position is in no way unique. I am citing from Beltrametti and Cassinelli: "The Logic of Quantum Mechanics" page 175:

"The possibilities of inventing contextual theories has been further illuminated by a sharp theorem of Gudder according to which the (L, S) pair, even under hypotheses slightly weaker than the ones we have assumed, always admits a contextual hidden-variable theory. Needless to say, this result does not secure for these contextual theories the status of physical theories. [...] presently, no clear empirical evidence at all has been found in favor of a hidden-variable theory"

If we are now talking about beables, this is within my area of expertise and beables as originally envisioned by Bell did not pan out. The only serious proposal for them that I am aware of is this paper: http://arxiv.org/pdf/quant-ph/9711009v1.pdf and in there please see note 11 on page 11: "the abstract counterpart of a Segalgebra is called a Jordan-Lie-Banach algebra". Now I do know Jordan-Lie-Banach algebras very well and in the finite dimensional case the "Segalgebras" are not independent of C* algebras and hence beables ARE observables. The infinite dimensional case is still open and this is one of my research topics. To my knowledge there is no mathematical consistent proposal where beables are not observables. If you do know of such a proposal, please do point it out to me: I am very interested to study it.

5. Please take a look at the following article:

http://phys.org/news/2014-02-team-limits-famous-quantum-principle.html

Does it have any consequences on determinism? And what about free will? And what about standard quantum mechanical interpretations (eg. Copenhagen and Many world)?

Regards

D.

6. Dear D,

The web reference you mentioned can be better understood from the corresponding archive paper link: http://arxiv.org/pdf/1308.5688v2.pdf This is only an experimental result which (of course) confirms quantum mechanics, but it is pushing the limits on measurement accuracy. As such, I am sorry to disappoint, but it has no relevance to determinism, free will or QM interpretations. Except free will I did touch on those topics in prior posts, please browse and read the prior posts.

Bell theorem shows that local realism is not a valid explanation of quantum mechanical effects: some people may say that there is no causal explanation possible for quantum correlations, but this is not accurate. Free will is a big open problem with no universally accepted solution. Free will touches on core issues in physics and QM: is randomness fundamental?, is QM universal?, what is the relationship between between QM, information, entropy, and time? is there superdeterminism? So far there are no known experiments distinguishing between QM interpretations.

According to that article, there's an "exceptional quantum state fidelities of up to 0.999 98(6)". In sum, doesn't it mean that the uncertainty of uncertainty principle is ruled out?

Moreover, from what I read on an important paper:

http://arxiv.org/abs/1308.5290

"Quantum theory is a well-defined local theory with a clear interpretation. No "measurement problem" or any other foundational matters are waiting to be settled"

Also, the paper states that there's not wave function collapse, quantum evolution is not reversible, there's not half alive hald dead cat paradox, no instant action at distance. Thus, why so many scientists still debate about such issues?

What I fear is that today science is broken, after all. Trying to find an interpretation in QM could be a waste of time since there is not a clear empirical evidence over many issues. What is your opinion?

Best regards

D.

8. "In sum, doesn't it mean that the uncertainty of uncertainty principle is ruled out?" Nope. Fidelity in that context means the accuracy of the measurement. It is like to how many accuracy digits one measures the speed of light or example.

On http://arxiv.org/abs/1308.5290, I am not buying it and I don't get how it got past the editors. There IS an open measurement problem in QM, and while I did not read the paper in detail, I skimmed through the proposed answer of the paper:

"In summary, then, the alleged “measurement problem”
does not exist as a problem of quantum theory. Those who
want to pursue the question Why are there events? must

Now the "seek the answer elsewhere' statement does not fit with the overall rosy picture the paper is painting, does it? So all is fine and dandy if we will just conveniently close our eyes and pretend the problem does not exist.

On various statements:
"there's not wave function collapse" - I agree with this, but not everyone does. There is a school of though (quantum bayesianism) which understands the collapse as a mere information update. My contention with that stems from my research of recovering QM from fundamental principles. Collapse means that unitary evolution is broken, and this makes the entire mathematical formalism inconsistent.

"quantum evolution is not reversible". Now I have to see this in the paper, because it means that the collapse is real and contradicts the point above. If collapse is real, then evolution is not reversible.

"there's not half alive hald dead cat paradox" - everyone agrees with this, but why this is not a paradox varies based on the interpretation.

"no instant action at distance" The jury is out on this too. Naive action at a distance (the one able to send signals) is not possible but in QM there is a thing called "quantum steering". Now quantum steering has no relativistic description possible and can be understood as peculiar quantum correlations. In realistic interpretations of QM this is a real effect, in epistemic interpretations (quantum bayesianism) it is not.

"Thus, why so many scientists still debate about such issues?"-because there are not settled.

Look at this: I3 and write it on a board. What did you write?

Is it 13? ...I2 I3 I4...
or is it B?
A
B
C

A
I2 I3 I4
B

I3 is like QM. 1213 14 A,B,C are like the interpretations (contexts). All interpretations are consistent and experiments cannot distinguish them. However there is a way out and science is not broken. Lorenz transformation were just as contentions 100 years ago as QM is today until special theory of relativity derived it from very simple physical principles. My research is to do the same for QM. Then the debate on interpretation will be over.

9. Dear D,

I finished reading the paper. The paper makes the following claims:

– Yes, quantum theory well deﬁned.
– Yes, quantum theory has a clear
interpretation.
– Yes, quantum theory is a local theory.
– No, quantum evolution is not reversible.
– No, wave functions do not collapse; you
– No, there is no instant action at a distance.
– Heisenberg’s cut is where you put it.
– No, Schr¨odinger’s cat is not half dead and
half alive.
– No, there is no “measurement problem.”

Here is my take on them:

>quantum theory well deﬁned. -- I AGREE
>quantum theory has a clear interpretation.- DISAGREE. However I am working on my own interpretation stemming from quantum reconstruction research. This will supersede all other interpretations
>quantum theory is a local theory. DISAGREE. QM violates Bell-locality.
>quantum evolution is not reversible. DISAGREE: there is the quantum eraser experiment (http://en.wikipedia.org/wiki/Quantum_eraser_experiment) and other results from quantum information
wave functions do not collapse; you reduce your state. I AGREE
>there is no instant action at a distance. - I DON'T KNOW in terms of quantum steering
>Heisenberg’s cut is where you put it. I AGREE
>Schr¨odinger’s cat is not half dead and half alive. I AGREE
>there is no “measurement problem.” I DISAGREE the problem is alive and well.

10. Thank you. Very eye-opening explanation.

Although I'm not as much confident as you with QM, I read that paper carefully and, honestly, I don't think that the answer given by the authors about those issues is just "let's close our eyes because it's all good". Nevertheless, I appreciate very much your answer because I've no doubt that it's given by a real super expert in the field of QM.

Finally, according to your interpetation, since there's not both wave function collapse and cat paradox, I'd like to ask which is the interpretation of your school of though about the famous Young double slit experiment.

Best luck with your wonderful work.