# How is this expression for a GHZ state obtained in the nature paper by Pan et al. (2000)?

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?

• web.physics.ucsb.edu/~quopt/ghz.pdf Jun 19, 2021 at 19:54
• 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? Jun 19, 2021 at 22:02
• Yes, that was my question. How can I substitute into 1 when equations 2 and 3 are V' and H' ??? Jun 19, 2021 at 22:05
• Do you mean expressing H and V in terms of H' and V' from 2 and 3 and then inserting them into 1?? Jun 19, 2021 at 22:07

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 $$(4)$$.

• Thanks a lot!!!! Jun 20, 2021 at 8:58
• 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. Jun 20, 2021 at 12:34