by **gill1109** » Sat Mar 06, 2021 10:26 pm

Curiosity wrote:gill1109 wrote: Yes, what he says is correct, but it is empty. There are no other local hidden variables models which reproduce the singlet correlations than models which can be cast in the form A (a, lambda) and B (b, lambda). Determinism is indeed irrelevant.

A local hidden variables model would (a) be local and (b) satisfy counterfactual definiteness. One could create a probability space with random variables A_a and B_b, for all directions a and b, such that A_a is the outcome which Alice would see if she chose setting a. Now define lambda = the pair of functions (A_a, B_b; a, b directions). Nature chooses a 0/1 valued function A_(.), and a function B_(.). The experimenter chooses settings a, b. The experimenter observes outcomes A_(a) and B_(b).

The no-conspiracy assumption says that nature and the experimenter are not constrained by one another.

I am not sure I understand. Are you saying your model includes the case when we do not assume deterministic functions A(.) but only probabilistic functions P_A(.)?

I am not saying that, but it is true.

I'm saying that a local hidden variables model, including a stochastic local hidden variables model, allows the creation of a classical probability space (Omega, F, P) with random variables X_a, Y_b defined on it. Thus X_a is a function of omega in Omega. If you like, you can think of omega as being lambda. But if you like, you can also think of the set of all functions {X_a(omega), Y_b(omega) : a, b directions} which you find when you let omega range through Omega.

If we are just interested in the probability distribution of a random variable X we can forget about the underlying probability space Omega and just take the new probability space IR (the real line) with the induced probability measure P_X on it defined as follows: for Borel subsets B of IR, P_X(B) = P(X in B) := P({omega: X(omega) in B}).

This is basic stuff in the complete standard measure theoretic foundation of probability. The theory built by Borel and Lebesgue and completed by Kolmogorov. And standard terminology from the theory of stochastic processes. We can think of all the possible outcomes of Alice's measuring spin in direction a as a random function, with time variable a. X_a(omega) as a varies, for fixed omega, is called the sample path of the stochastic process. Of course we do not observe that sample path. Alice chooses a direction a and gets to see one point on that random graph, namely X_a(omega).

[quote="Curiosity"][quote="gill1109"] Yes, what he says is correct, but it is empty. There are no other local hidden variables models which reproduce the singlet correlations than models which can be cast in the form A (a, lambda) and B (b, lambda). Determinism is indeed irrelevant.

A local hidden variables model would (a) be local and (b) satisfy counterfactual definiteness. One could create a probability space with random variables A_a and B_b, for all directions a and b, such that A_a is the outcome which Alice would see if she chose setting a. Now define lambda = the pair of functions (A_a, B_b; a, b directions). Nature chooses a 0/1 valued function A_(.), and a function B_(.). The experimenter chooses settings a, b. The experimenter observes outcomes A_(a) and B_(b).

The no-conspiracy assumption says that nature and the experimenter are not constrained by one another.[/quote]

I am not sure I understand. Are you saying your model includes the case when we do not assume deterministic functions A(.) but only probabilistic functions P_A(.)?[/quote]

I am not saying that, but it is true.

I'm saying that a local hidden variables model, including a stochastic local hidden variables model, allows the creation of a classical probability space (Omega, F, P) with random variables X_a, Y_b defined on it. Thus X_a is a function of omega in Omega. If you like, you can think of omega as being lambda. But if you like, you can also think of the set of all functions {X_a(omega), Y_b(omega) : a, b directions} which you find when you let omega range through Omega.

If we are just interested in the probability distribution of a random variable X we can forget about the underlying probability space Omega and just take the new probability space IR (the real line) with the induced probability measure P_X on it defined as follows: for Borel subsets B of IR, P_X(B) = P(X in B) := P({omega: X(omega) in B}).

This is basic stuff in the complete standard measure theoretic foundation of probability. The theory built by Borel and Lebesgue and completed by Kolmogorov. And standard terminology from the theory of stochastic processes. We can think of all the possible outcomes of Alice's measuring spin in direction a as a random function, with time variable a. X_a(omega) as a varies, for fixed omega, is called the sample path of the stochastic process. Of course we do not observe that sample path. Alice chooses a direction a and gets to see one point on that random graph, namely X_a(omega).