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I was reading about a Bell’s inequality experiment from the paper https://arxiv.org/abs/quant-ph/0205171.

In the paper it gives a set of recorded values:

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I understand how they got the value for S, but I am having trouble understanding how to use this formula to find the error:

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The final answer should be: $S = 2.307 ± 0.035$

What should $N_i(\partial S/\partial N_i)^2$ be?

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  • $\begingroup$ Eqs 21 and 25 tell you how to compute $S$ in terms of $N_i$, and the $N_i$ are tabulated in the second-to-last column of Table I. To get $\partial S/\partial N_i$, you just do a partial derivative of the expression for $S$. As the paper says, there will be a lot of terms. Do you have a specific step you are stuck on? $\endgroup$
    – Andrew
    Commented Nov 10, 2022 at 0:40
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    $\begingroup$ @kcoding Your last edit accidentally erased most of your question. $\endgroup$
    – David Bailey
    Commented Nov 10, 2022 at 15:40

1 Answer 1

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Rewriting Equation 21 in different notation,

\begin{equation} S = E(N_1,N_2,N_3,N_4) - E(N_5, N_6, N_7, N_8) + E(N_9, N_{10}, N_{11}, N_{12}) - E(N_{13}, N_{14}, N_{15}, N_{16}) \end{equation} where (rewriting Equation 25 in different notation) \begin{equation} E(N_i, N_j, N_k, N_l) = \frac{N_i + N_j - N_k - N_l}{N_i + N_j + N_k + N_l} \end{equation} So for example \begin{equation} \frac{\partial S}{\partial N_1} = \frac{1}{N_1 + N_2 + N_3 + N_4} - \frac{N_1+N_2-N_3-N_4}{\left(N_1 + N_2 + N_3 + N_4\right)^2} \end{equation}

WARNING The above is just meant to show you how the partial derivative would work. To actually figure out what entries in Table I correspond to $N_1, N_2$, etc, you need to be careful about whether the pairs of angles should be perpendicular or not, or whether the angles $a, a', b, b'$ are being used, following the original notation in Eqs 21 and 25 carefully.

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  • $\begingroup$ Thank you. Is the $N_1,N_2,N_3,N_4$ in the final equation you wrote the uncertainity in each values of $N_1,N_2,N_3,N_4$ or are they just the values $\endgroup$
    – Jack Jack
    Commented Nov 10, 2022 at 12:38
  • $\begingroup$ @kcoding Just the values. As an aside, in the last equation in your question, you see that the uncertainty is related to the value by $\sigma_{N_i}^2=N_i$, which follows from Poisson statistics. $\endgroup$
    – Andrew
    Commented Nov 10, 2022 at 13:12
  • $\begingroup$ wolframalpha.com/… $\endgroup$
    – Andrew
    Commented Nov 10, 2022 at 13:38

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