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Can someone tell me how the authors of the paper "Experimental test of quantum nonlocality" (Nature link to abstract) have rewritten their equation 1 in terms of equation 2 and 3?

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  • $\begingroup$ web.physics.ucsb.edu/~quopt/ghz.pdf $\endgroup$ Jun 19 at 19:54
  • $\begingroup$ To clarify: are you asking how $(4)$ comes from $(1)$, $(2)$ and $(3)$? If so, then you can probably just substitute inverted $(2)$ and inverted $(3)$ into $(1)$... If not, then could you clarify what the question is? $\endgroup$ Jun 19 at 22:02
  • $\begingroup$ Yes, that was my question. How can I substitute into 1 when equations 2 and 3 are V' and H' ??? $\endgroup$ Jun 19 at 22:05
  • $\begingroup$ Do you mean expressing H and V in terms of H' and V' from 2 and 3 and then inserting them into 1?? $\endgroup$ Jun 19 at 22:07
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First, in order to express $|H'\rangle$, $|V'\rangle$, $|R\rangle$ and $|L\rangle$ in terms of $|H\rangle$ and $|V\rangle$, add and subtract the pair of equations $(2)$ and add and subtract the pair $(3)$, to get

$$ |H'\rangle + |V'\rangle = \sqrt2|H\rangle \\ |H'\rangle - |V'\rangle = \sqrt2|V\rangle \\ |R\rangle + |L\rangle = \sqrt2|H\rangle \\ |R\rangle - |L\rangle = i\sqrt2|V\rangle $$

which means that

$$ |H\rangle = \frac{1}{\sqrt{2}}\left(|H'\rangle + |V'\rangle\right) = \frac{1}{\sqrt{2}}\left(|R\rangle + |L\rangle\right) \\ |V\rangle = \frac{1}{\sqrt{2}}\left(|H'\rangle - |V'\rangle\right) = \frac{1}{i\sqrt{2}}\left(|R\rangle - |L\rangle\right). $$

Substituting the last equations into $(1)$ according to the choices of polarization in text, we obtain

$$ \begin{align} |\Psi\rangle &= \frac{|H\rangle|H\rangle|H\rangle + |V\rangle|V\rangle|V\rangle}{\sqrt{2}} \\ & = \frac{(|R\rangle + |L\rangle)(|R\rangle + |L\rangle)(|H'\rangle + |V'\rangle)-(|R\rangle - |L\rangle)(|R\rangle - |L\rangle)(|H'\rangle - |V'\rangle)}{4}. \end{align} $$

Notice that the only terms that do not cancel are those with odd number of kets in $\{|L\rangle,|V'\rangle\}$, i.e.

$$ |\Psi\rangle = \frac{1}{2}\left(|L\rangle|R\rangle|H'\rangle + |R\rangle|L\rangle|H'\rangle + |R\rangle|R\rangle|V'\rangle + |L\rangle|L\rangle|V'\rangle\right) $$

in agreement with $(4)$.

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  • $\begingroup$ Thanks a lot!!!! $\endgroup$ Jun 20 at 8:58
  • 2
    $\begingroup$ Hi @HaxhiPantina and welcome to QCSE! If an answer solved your doubts, consider accepting it by clicking the check mark. This helps keep focus on questions that haven’t been solved. $\endgroup$
    – epelaaez
    Jun 20 at 12:34

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