Joy Christian wrote:
You can find the above comments from Einstein on page 89 of the collection of papers by Shimony, cited as Ref. [6] in my paper.
Please give me the reference to the original paper by Shimony. The Cambridge University Press book you refer to is only available in hard-cover, costs maybe 100 Euro and will take time to get hold of.
In the meantime, I can only quote Bohr: "How wonderful that we have met with a paradox. Now we have some hope of making progress.".
I am certain, for many reasons, that Bell's theorem is not homologous to von Neumann's failed proof. We must be careful to distinguish quantum dogma (the pronouncement "thou shalt not..." uttered by the high priests of quantum physics) from the bare facts predicted by core quantum theory, and moreover confirmed experimentally. Secondly, when discussing hidden variables models, we must remember that a hidden variables theory as envisaged by Einstein would be a classical-like theory which underlay quantum theory, thus making the same (or at least, experimentally indistinguishable) predictions as quantum theory. So if quantum theory predicts that the sum of the expectation values of two observables is equal to the expectation value of a third, then any hidden variable theory which actually does reproduce (accurately enough) quantum theory, must satisfy the same rule (or at least, satisfy it, to a close approximation).
It seems to me that Einstein's argument is wrong. The linearity does have to hold. It is something else which goes wrong. We have made an interesting discovery. Our idol was not always right...
So far you are just quoting authority to support your point of view. You are not responding to my argument. A true scientist does not accept things on the authority of great scientists (or great religious authorities) of the past. A true scientist accepts things on the basis of logical argument and empirical observation.
Added a short time later:
Einstein says that the linearity of expectation values supposed to hold in quantum theory need only hold under states which exist in quantum theory. That is true, but that is not the issue here. Moreover, he is careless in his words. The linearity of what? We must distinguish carefully between objects existing in certain mathematical theories. The theories are different. They correspond in some way to things observed in nature but they are not the same as those things. This is a typical mistake of physicists, to get reality mixed up with the abstract (formal, mathematical) objects in the language (mathematics) they use to describe reality.
In a hidden variable theory, the true state of nature is described by further, presently un-observable, variables. Einstein says that when you further condition on those variables, the additivity between the things which underly those observables need no longer hold. The point is, the quantum observable is not the same thing as the values observed when the observable is measured. In my spin example, sigma_x + sigma_z = sqrt 2 sigma_u is an identity relating three self-adjoint observables. Suppose we came up with a hidden variables theory for the three spins, which stated that on measuring any of those observables, we merely observe the value of a pre-existing random variable (ie a function of the true underlying state of the system). Then there would exist three random variables on a single probability space such that the probability distribution of any one of the three was the probability distribution of the outcomes of measuring the corresponding observable. Call these three random variables X, Z, and U. The three random variables would have to take the values +/-1 only with probabilities which can be deduced from their three expectation values trace rho sigma_x etc.
It would have to be the case that E(X) + E(Z) = E(sqrt 2 . U). As I already said, please check yourself that trace(rho sigma_x) + trace(rho sigma_z) = sqrt 2 trace(rho sigma_u). The linearity does hold, both in the hidden variables theory and in quantum theory! The problem is that the sum of two random variables taking the values +/-1 is a random variable taking the values -2, 0, and +2; while sqrt 2 U is a random variable taking the values +/- sqrt 2.