I was wrong and Lubos Motl was right
but quantum correlations are not like Bertlmann's socks
It finally happened. I was too careless and I stupidly challenged Lubos to "Show me a non-factorizable state in CM for spatially separated physical systems!!!"
And he indeed presented one:
This is obscurely written but it is ultimately correct. Glup, glup, glup, he sank my battleship, bruised my ego, won the battle, but not the war :) I will attempt to show that quantum correlations are not like Bertlmann's socks correlations because quantum correlations depend in an essential way on the observer. First let me clarify Lubos example and why he is right in his example.
First what is this business of Bertlmann's socks? This came from a famous Bell paper:
Mr Bertlmann is a real person who uses to wear socks of different colors. The moment you see that one of his socks is pink you know that his other sock is not pink. (Lubos is using red and green) and there is a correlation between the sock colors. In general, you learn from statistic 101 that a joint probability \(P(A, B)\) factorizes: \(P(A,B)=P(A)P(B)\) if the two probabilities are independent. Therefore it was blatantly stupid on my part to ask Lubos for a non-factorizable example. But I had something else in mind for which I carelessly skipped the essential details and now I will explain it properly.
The Bell theorem factorization condition is not on independent probabilities but on residual probabilities:
\(P(A,B, \lambda) = P(A, \lambda) P(B, \lambda)\)
In the case of Bertlmann's socks there are no residual probabilities! But what does this mean?
Correlations can be generated due to physical interaction between two systems or because they share a common cause. One way people explain this is by inventing silly games where two players agree in advance on a strategy but they are not allowed to communicate during the actual game. For example consider this: Alice and Bob each have a coin and they flip it heads or tails. Then they both have to guess the result of each other and they win the game when successful. If they guess randomly the best odds of wining the game is 25% of the time. However there are two strategies they can agree beforehand which increases their odds of winning to 50%. (Can you guess what those strategies might be?). Strategies, common causes and interactions, outcome filtering, all generate correlations and ruin the factorization condition.
But quantum mechanics is probabilistic and even if you account for all interactions the outcome is still random. After accounting for all those factors in the form of a generic variable \(\lambda\), the remaining probability is called a residual probability. Lambda can be a variable, or a set of variables of an unspecified format. The point is that after accounting for all common causes, if the physical systems A and B are spatially separated, they cannot communicate and in the quantum case it seems reasonable to demand \(P(A,B, \lambda) = P(A, \lambda) P(B, \lambda)\).
But such a factorization is at odds with both quantum mechanics predictions, and experimental results. This is the basis on which people in foundations of quantum mechanics call nature nonlocal.
I think I know how Lubos may attack this. I bet he will say that \(\lambda\) is basically a hidden variable and quantum mechanics does not admit hidden variables. This line of argument is faulty. Carefully read (several times) Bell's paper from the link above and you see that \(\lambda\) represents the coding of the usual way correlations are accounted by this common \(\lambda\) which is present in all 3 factors.
Let me spell out better Bell's argument following a well written paper by Bernard d'Espagnat. Bell considers the singlet state and in there you have Alice and Bob in two spatially separated labs measuring the spins on directions a, and b and obtaining the outcomes A, and B respectively, Let \(\lambda\) represent a common source of the correlation between A and B. Then one can write the standard rule of statistics: \(P(M,N) = P(M|N)P(N)\) like this:
\(P(A,B|a,b,\lambda) = P(A|a,b,B,\lambda)P(B|a,b,\lambda)\)
then because what happens at Alice's side does not depend on what happens on Bob side and the other way around:
\(P(A|a,b,B,\lambda) = P(A|a, \lambda)\)
\(P(B|a,b,\lambda) = P(B|b, \lambda)\)
\(P(A,B|a,b,\lambda) = P(A|a, \lambda) P(B|b,\lambda)\)
From this the usual Bell theorem follows and disagreement with experiment is used to point out that what happens at Alice's side does depend on what happens on Bob side. In other words, nonlocality.
I disagree (with good arguments) with several points of view:
- I disagree with Lubos that nature does not follow the logic of projectors and follows the Boolean logic instead. (measurements project on a Hilbert subspace and quantum OR and quantum NOT are different than their classical counterparts)
- I disagree with Tim Maudlin who best defends the point that Bell proved nonlocality based on the argument above (Quantum mechanics is contextual and because of that: \(P(A|a,b,B,\lambda) \ne P(A|a, \lambda)\) . As such Bell's factorization condition is not justified. Only if you think in terms of classical Boolean logic avoiding contextuality the nonlocality conclusion is inescapable. )
- I disagree with Lubos that quantum correlations are like Bertlmann's socks. To eliminate thinking about lambda as a hidden variable, picture it as fixing all conceivable sources of correlations (Bertlmann's socks type or any other type). Now add locality independence, Boolean logic, and you get correlations at odds with experiments. Pick your poison: give up locality or give up Boolean logic. The one to give up is Boolean logic. Unlike Bertlmann's socks correlations, quantum correlations depend in an essential way on the observer.
- I disagree with giving up on locality and that the Bohmian position represents a valid description of nature (what happens with he quantum potential of a particle after the particle encounters its antiparticle? Both vanish or not vanish result in predictions incompatible with observations)
Bell himself provided 4 possible explanations of quantum mechanics' correlations:
-quantum mechanics is wrong sometimes
-superdeterminism (lack of free will)
-faster than light causal influences
My take on this is that quantum mechanics is the complete and correct description of nature, there is free will, there are no faster than light causal influences, realism is incorrect, and what people call nonlocality is actually a manifestation of contextuality because the observer (but not consciousness) does play an active role in generating the experimental outcome. This active role happens even when parts of the composed system are out of causal reach because quantum mechanics is blind to space-time separations.