Quantum math on GHZ state equation

[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.
Please help if you can!

 

[eljose79]

Thank you for your interest in recent tests of quantum nonlocality and for bringing up this paper by Pan, Bouwmeester, Daniell, Weinfurter, and Zeilinger. The key equation, |Ψ⟩=1/√2 (|H⟩_1 |H⟩_2 |H⟩_3 + |V⟩_1 |V⟩_2 |V⟩_3), represents a 3-photon GHZ state, where |H⟩ and |V⟩ represent horizontal and vertical polarizations, respectively.

To understand how equation (4) was derived, we need to look at the optical devices introduced in the experiments. These devices are used to manipulate the polarizations of the photons, and the equations (2) and (3) show the effect of these devices on the polarizations.

In equation (2), the pair of states |H/⟩ and |V/⟩ represent right-handed and left-handed circular polarizations, respectively. These are obtained by adding the |H⟩ and |V⟩ states in equal proportions and dividing by √2. Similarly, in equation (3), the pair of states |ʘR⟩ and |ʘL⟩ represent right and left circular polarizations with a phase difference of π/2. These are obtained by adding the |H⟩ and i|V⟩ states (for |ʘR⟩) or the |H⟩ and -i|V⟩ states (for |ʘL⟩) in equal proportions and dividing by √2.

Now, to get to equation (4), we need to substitute these pairs of states into the original equation (1). This is where the vector algebra comes in. We can think of the states as vectors in a 3-dimensional space, with the horizontal polarization represented by the x-axis, vertical polarization by the y-axis, and circular polarizations by the z-axis.

So, for example, the state |H/⟩ can be represented as a vector (1/√2, 1/√2, 0), and the state |V/⟩ as (1/√2, -1/√2, 0). Similarly, |ʘR⟩ can be represented as (1/√2, 0, i/√2) and |ʘL⟩ as (1/√2, 0, -i