# Quantum math on GHZ state equation

1. Feb 23, 2012

### ljagerman

Interested in recent tests of quantum nonlocality in 3-photon GHZ entanglement.
Looking at key paper by Pan, Bouwmeester, Daniell, Weinfurter & Zeilinger in Nature, 403, 2 Feb 2000, p. 515-518.
Key equation is for entangled 3-photon GHZ state; their equation (1):
|Ψ⟩=1/√2 (|H⟩_1 |H⟩_2 |H⟩_3 + |V⟩_1 |V⟩_2 |V⟩_3); no problem here.
Then the experiments introduce optical devices to rotate polarization and induce circluar polarization; equations here are (their (2) and (3): [ʘR=right-handed circular polarization, etc.]
Pair (2): |H/⟩ =1/√2 |H⟩+|V⟩) and |V/⟩ =1/√2 |H⟩+|V⟩)
Pair (3): |ʘR⟩ =1/√2 |H⟩+i|V⟩) and |ʘL⟩ =1/√2 |H⟩-i|V⟩)
This looks like simple vector trig in a unit circle.
Now they solve (1) above with (2) and (3), and they arrive at their equation (4):
|Ψ⟩=1/2 (|ʘR⟩_1 |ʘL⟩_2 |H/⟩_3 +
|ʘL⟩_1 |ʘR⟩_2 |H/⟩_3 +
|ʘR⟩_1 |ʘR⟩_2 |V/⟩_3 +
|ʘL⟩_1 |ʘL⟩_2 |V/⟩_3).
(I spread it out into 4 lines to make it easier to read.)
My question: How did they get to (4)? What vector-algebraic (? or other) steps did they use? I'm guessing it's some form of substitution.
I know a little about Dirac notation, wave functions, polarization, etc, but not enough to see how equation (4) was derived.