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.

 

[Point Conception]

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).