Can Non Realism = Non Deterministic Hidden Variable Theory

[stevendaryl]

So λ (dependent on θ₂ - θ₁) in this case involves not only all information of past variable but also involves the physical interactions during measurement: λ_t0 (ontic) , settings a and b, λ _t1 (observed)

Two things that I don't understand about that one sentence: First, \lambda is supposed to be something that is "set" in the backwards light-cone of the two measurements--in other words, at the moment the twin pair is produced. \theta_2 - \theta_1 is a fact about the measurement process, and is definitely NOT in the backwards light-cone. That quantity can be changed in-flight, right before the detection event.

The second thing I don't understand is how \theta_2 - \theta_1 is a local quantity. It depends on two distant quantities. By definition, I think that would be nonlocal.

 

[morrobay]

Two things that I don't understand about that one sentence: First, \lambda is supposed to be something that is "set" in the backwards light-cone of the two measurements--in other words, at the moment the twin pair is produced. \theta_2 - \theta_1 is a fact about the measurement process, and is definitely NOT in the backwards light-cone. That quantity can be changed in-flight, right before the detection event.

The second thing I don't understand is how \theta_2 - \theta_1 is a local quantity. It depends on two distant quantities. By definition, I think that would be nonlocal.

When you say that λ is "set" in past light cone, produced at entanglement of pair, are you implying that λ does not interact with observable during measurement process ?
Ie ( λt₀ + ontic unmeasured particle ). ---> interactions with detector setting a or b at A or B ---> ( λ_t1 + observed particle measurement.)
Maybe someone can elaborate on exactly how λ and particle interact physically starting from moment pair is produced ,
during measurement interaction , to observed outcome.
And I redefined contextual hidden variable to depend on experimental setting ,θ, at A or B for locality .And as you say detector angle can be changed in flight. So whether λ is a function, outcome ± 1 or outcome is stochastic I cannot see how counter- factual definiteness could apply.
So again is a local , non realistic hidden variable theory possible from the above : That can have perfect correlations when detectors are aligned and also predict inequality violations, S_qm = 2√2 when detectors at A and B are not aligned ?

 

[stevendaryl]

When you say that λ is "set" in past light cone, produced at entanglement of pair, are you implying that λ does not interact with observable during measurement process ?

The outcome at each detector is assumed to be a function of the variable \lambda, which is set at the moment of pair creation, and the detector setting.

There is a joint probability function, P(\theta_1, \theta_2, A, B) which is the probability that Alice gets outcome A, and Bob gets outcome B, given that Alice's device has setting \theta_1 and Bob's device has setting \theta_2.

To explain this joint probability distribution in terms of local hidden variables would be to write it in the form:

P(\theta_1, \theta_2, A, B) = \sum_\lambda P(\lambda) P(\theta_1, A, \lambda) P(\theta_2, B, \lambda)

where

• P(\lambda) is the probability that the hidden variable has value \lambda
• P(\theta_1, A, \lambda) is the probability that Alice gets outcome A given the hidden variable has value \lambda and her device has setting \theta_1
• P(\theta_2, B, \lambda) is the probability that Bob gets outcome B given the hidden variable has value \lambda and his device has setting \theta_2

So, yes, the outcome for Alice is assumed to depend on some kind of interaction between \lambda and \theta_1, and the outcome for Bob depends on some kind of interaction between \lambda and \theta_2. But \lambda has nothing to do with \theta_2 - \theta_1.

 

[stevendaryl]

So whether λ is a function, outcome ± 1 or outcome is stochastic I cannot see how counter- factual definiteness could apply.

We assume a probability distribution of the form:

P(\theta_1, \theta_2, A, B) = \sum_\lambda P_A(\lambda) P_B(\theta_1, A, \lambda) P(\theta_2, B, \lambda)

Now, we use the fact that if \theta_1 = \theta_2, then the correlation (or anti-correlation) is perfect. So we have, for perfect anti-correlation:

P(\theta, \theta, A, A) = \sum_\lambda P(\lambda) P_A(\theta, A, \lambda) P_B(\theta, A, \lambda) = 0

There is no way to have a sum of nonnegative terms add up to 0 unless each term is 0. So we conclude:

P_A(\theta, A, \lambda) P_B(\theta, A, \lambda) = 0

(for all values of \lambda with nonzero probability).

This implies

P_A(\theta, A, \lambda) = 0 or P_B(\theta, B, \lambda) = 0

This is true for every value of \lambda and \theta. That means that for any value of \lambda, there is some angle \theta such that either it is impossible for Alice to get result A at that angle, or it is impossible for Bob to get result A at that angle. So, if Alice gets result A at some angle \theta, then it is DEFINITE that Bob cannot get result A at that angle. That's contrafactual definiteness. It follows from the assumption of local hidden variables and the fact of perfect anti-correlation (or correlation).