The CH inequality again assumes the hidden variables recur under different configurations. When the expressions are written taking into account the independence of each configuration the upper bound increases to 1. Experiments that violate the upper bound of 0 do not violate the upper bound of 1 and hence a local model is not ruled out by 0 violation.

Clauser [5] in his “Bells theorem: experimental tests and implications” writes

*Following our discussion of §3.3, we assumed that, given λ*, *a and b, the probabilities p*_{1}(*λ*, *a*) *and p*_{2}(*λ*, *b*) *are independent. Thus we write the probabilities of detecting both components as*
$$\begin{array}{}{p}_{12}(\lambda ,a,b)={p}_{1}(\lambda ,a){p}_{2}(\lambda ,b)\end{array}$$(3.15)

*The ensemble average probabilities of equations (3.14) are then given by*:
$$\begin{array}{rl}{p}_{1}(a)\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& ={\displaystyle \int {}_{\mathit{\Lambda}}{p}_{1}\left(\lambda ,a\right)d\rho}\\ {p}_{2}(b)\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& ={\displaystyle \int {}_{\mathit{\Lambda}}{p}_{2}\left(\lambda ,b\right)d\rho}\\ {p}_{12}(a,b)\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& ={\displaystyle \int {}_{\mathit{\Lambda}}{p}_{12}\left(\lambda ,a,b\right)d\rho}\end{array}$$(3.16)

*To proceed, CH introduce the following lemma, the proof of which may be found in their paper: if x*, *x*′, *y*, *y*′, *X*, *Y are real numbers such that* 0 ≤ *x*, *x*′ ≤ *X and* 0 ≤ *y*, *y*′ ≤ *Y, then the inequality*
$$\begin{array}{}-XY\le xy-x{y}^{{}^{\prime}}+{x}^{{}^{\prime}}y+{x}^{{}^{\prime}}{y}^{{}^{\prime}}-Y{x}^{{}^{\prime}}-Xy\le 0\end{array}$$(3.17)

*holds. Inequality (3.17) and equation (3.15) yield*:
$$\begin{array}{}-1\le {p}_{12}(\lambda ,a,b)\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{p}_{12}(\lambda ,a,{b}^{{}^{\prime}})\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{p}_{12}(\lambda ,{a}^{{}^{\prime}},b)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+{p}_{12}(\lambda ,{a}^{{}^{\prime}},{b}^{{}^{\prime}})-{p}_{1}(\lambda ,{a}^{{}^{\prime}})-{p}_{2}(\lambda ,b)\le 0\end{array}$$(3.18)

*Integrating inequality (3.18) over λ with distribution ρ, and using equation (3.16), one obtains the result*:
$$\begin{array}{}-1\le {p}_{12}(a,b)-{p}_{12}(a,{b}^{{}^{\prime}})+{p}_{12}({a}^{{}^{\prime}},b)+{p}_{12}({a}^{{}^{\prime}},{b}^{{}^{\prime}})\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-{p}_{1}({a}^{{}^{\prime}})-{p}_{2}(b)\le 0\end{array}$$(3)

Implicit in the derivation of (3) is the assumption the same *λ* can appear with different configurations (*a*, *b*), (*a*, *b*′), (*a*′, *b*), and (*a*′, *b*′). That assumption is false for nonrecurrent random variables. Each configuration has its own unique set of hidden variables. Hence equation (3.18) does not hold for those hidden variables. That equation lacks generality. If *λ* appears in *p*_{12}(*λ*, *a*, *b*) it does not appear in any of the other terms. For nonrecurrent hidden variables the context in (3.15) must be specified. The CH recurrent form (3.17) has 4 variables: *x*, *x*′, *y*, *y*′. The corresponding nonrecurrent form has 8 variables:
$$\begin{array}{}0\le {x}_{i}\le X,0\le {y}_{i}\le Y,i=1,2,3,4\end{array}$$

We repeat *(3.17)* with the *recurrent* “kernel” specified as
$$\begin{array}{}{K}_{R}=xy-x{y}^{{}^{\prime}}+{x}^{{}^{\prime}}y+{x}^{{}^{\prime}}{y}^{{}^{\prime}}-Y{x}^{{}^{\prime}}-Xy\end{array}$$

and then write the corresponding *nonrecurrent* “kernel”
$$\begin{array}{}{K}_{N}={x}_{1}{y}_{1}-{x}_{2}{y}_{2}+{x}_{3}{y}_{3}+{x}_{4}{y}_{4}-Y{x}_{3}-X{y}_{1}\end{array}$$

The choice of -*Yx*_{3} -*Xy*_{1} is arbitrary. They could just as well have been set to -*Yx*_{5} -*Xy*_{5} in a fifth experiment but we seek the least upper bound and assume the x and y are taken from the experiments as written. Doing so increases the constraints on the variables. Regroup the nonrecurrent kernel and write
$$\begin{array}{}{K}_{N}=\left({x}_{1}-X\right){y}_{1}-{x}_{2}{y}_{2}+{x}_{3}\left({y}_{3}-Y\right)+{x}_{4}{y}_{4}\end{array}$$

Each term in that expression is independent of every other term. As such the max of K_{N} is obtained by maximizing each term separately.
$$\begin{array}{}MAX\left({K}_{N}\right)=MAX\left(\left({x}_{1}-X\right){y}_{1}\right)+MAX\left(-{x}_{2}{y}_{2}\right)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+MAX\left({x}_{3}\left({y}_{3}-Y\right)\right)+MAX\left({x}_{4}{y}_{4}\right)\\ \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}MAX\left({K}_{N}\right)=0+0+0+XY=XY\end{array}$$

The minimum is likewise determined
$$\begin{array}{}[MIN\left({K}_{N}\right)=MIN\left(\left({x}_{1}-X\right){y}_{1}\right)+MIN\left(-{x}_{2}{y}_{2}\right)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+MIN\left({x}_{3}\left({y}_{3}-Y\right)\right)+MIN\left({x}_{4}{y}_{4}\right)\\ \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}MIN\left({K}_{N}\right)=-XY-XY-XY+0=-3XY\end{array}$$

Setting XY = 1 leads to
$$\begin{array}{}-3\le {K}_{N}\le 1\end{array}$$

as compared to
$$\begin{array}{}-1\le {K}_{R}\le 0\end{array}$$

The upper bound for experimental results is 1 and not 0. Hence any experiment that exceeds 0 does not imply nonlocality but rather that the hidden variables are nonrecurrent (and local).

A recent paper by Giustina et al. [9] purports to violate the CH-Eberhard upper bound of 0 and hence simultaneously close several loopholes. The nonrecurrent upper bound is 1 for CH and hence there is no difficulty explaining those experimental results with a local model. The CH derivation is invalidated by nonrecurrent hidden variables. In general the CH inequality is false.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.