(p.251) Appendix I A More General Common‐Cause Model for EPR
(p.251) Appendix I A More General Common‐Cause Model for EPR
The aim is to construct a model for the EPR arrangement that explicitly includes the apparatuses. The critical assumption in considering the apparatuses and their states is that the two measurements are supposed to be made simultaneously. Thus the state of the apparatus in one wing at the time of measurement can have no effect on the outcome in the other. Indeed, in the experiments of Aspect et al.^{1} the choice of direction is made at the very last instant, to ensure that no causal connection is possible. Some other assumptions will make the notation easier. Suppose for simplicity that in any complete cause in which states of the apparatus figure as a part, it is always the same apparatus states. Designate the set of states positively relevant to spin‐up outcomes along direction θ in the left‐wing apparatus, m_{L,θ}, those for the apparatus in the right wing, m_{R,θ}. Allowing initially all combinations of possible partial causes, set
What kinds of causal connection are appropriate for an EPR experiment? The usual assumptions are

1. The state of the measuring apparatus on the right (left) does not influence the outcome on the left (right). This is expressed formally in the model by setting
$${\hat{a}}_{L\text{,}\theta}(h\text{,}i\text{,}1\text{,}k)=0={\hat{a}}_{R\text{,}\theta \prime}(h\text{,}1\text{,}j\text{,}k)$$ 
2. The outcome on the right (left) does not affect the outcome on the left (right). So
$${\hat{a}}_{L\text{,}\theta}(h\text{,}i\text{,}j\text{,}1)=0={\hat{a}}_{R\text{,}\theta \prime}(h\text{,}i\text{,}j\text{,}1)$$ 
3. x _{1} is not a complete cause of either the outcome on the left nor of the one on the right:
$${\hat{a}}_{L\text{,}\theta}(1\text{,}0\text{,}0\text{,}0)=0={\hat{a}}_{R\text{,}\theta \prime}(1\text{,}0\text{,}0\text{,}0)$$
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4. The state of the apparatus on the left (right) is not a complete cause of the outcome on the left (right):
$${\hat{a}}_{L\text{,}\theta}(0\text{,}1\text{,}0\text{,}0)=0={\hat{a}}_{R\text{,}\theta \prime}(0\text{,}0\text{,}1\text{,}0)$$ 
5. x _{1} and m_{L, θ} are joint partial causes of x_{L, θ}; and x _{1} and m_{R,θ′}, of x _{R,θ′}:
$${\hat{a}}_{L\text{,}\theta}(1\text{,}1\text{,}0\text{,}0)=0={\hat{a}}_{R\text{,}\theta \prime}(1\text{,}0\text{,}1\text{,}0)$$Relabelling â_{L,θ} (1, 1, 0, 0) as â_{L,θ}; and â_{R,θ′} (1, 0, 1, 0) as â_{R,θ′}, the basic equations of the model will be:
$$\begin{array}{cc}\hfill & {x}_{L}(\theta )\equiv {\hat{a}}_{L\text{,}\theta}\xb7{x}_{1}\xb7{\hat{m}}_{L\text{,}\theta}\nu {u}_{L}\hfill \\ \multicolumn{1}{c}{}& {x}_{R}(\theta \prime )\equiv {\hat{a}}_{R\text{,}\theta \prime}\xb7{x}_{1}\xb7{\hat{m}}_{R\text{,}\theta \prime}\nu {u}_{R}\hfill \end{array}$$
Notes:
(1) A. Aspect, J. Dalibard, and G. Roger, ‘Experimental Tests of Bell's Inequalities Using Time Varying Analyzers’, Physical Review Letters, 49 (1982), 1804–7.