I'm reading through Valerio Scarani's "Six Quantum Pieces" and have hit an exercise which requires verification of an equality.

where the letters in the brackets indicate the photons.

How does one verify the above equality. The answer given in the book is

That is where I am stuck. I know the different bell-states, but entangle swapping photons that are entangled differently I can't figure out what the outcomes will be.

Apparently the right hand side is an expansion of |Psi+>{AB} |Phi+>{CD}, not of |Psi+>{DA} |Phi+>{BC}. Now I'm extra confused at how the author obtains
= Phi+(AB)Psi+(CD) + Phi-(AB)Psi-(CD) + Psi+(AB)Phi+(CD) + Psi-(AB)Phi-(CD)
from
|Psi+>{AB} |Phi+>{CD}, not of |Psi+>{DA} |Phi+>{BC}

Okay, so I expand the right hand side:
=(|H>|H>+|V>|V>(AB) |H>|V>+|V>|H>(CD)) + (|H>|H>-|V>|V>(AB) |H>|V>-|V>|H>(CD)) + (|H>|V>+|V>|H>(AB) |H>|H>+|V>|V>(CD)) + (|H>|V>-|V>|H>(AB) |H>|H>-|V>|V>(CD))

That done, I don't see how that is equivalent to |H>|V>+|V>|H>(DA) and |H>|H>+|V>|V>(BC).

If instead photons D&A are entangled as |H>|V> - |V>|H>, what impact does that have on the equality presented?

If the photons D&A and B&C are both in the bell-state |H>|H> + |V>|V> or |H>|H> - |V>|V>, would detecting photons A & B in the bell-state Phi+ mean photons C & D are in that bell-state also? Or if D&A are entangled as |H>|H> + |V>|V> and B&C are entangled as |H>|H> - |V>|V>, would we find photons C&D in the opposite bell-state that photons A & B are found in (i.e C&D in Psi-, A&B in Psi+)?

Do not use labels. Always write things in the same order A B C D i.e HVHH for A = H, B = V C = H and D = H.
It is easy for le right side (they are in the good order. For the left side you ave DA and BC terms, take Psi+(DA)Phi+(BC)
you have a HH term for BC, you write it and a HV for DA. You write V at the left and H at the right of the HH and you get VHHH for ABCD and so on for the four terms of the left side.

Here we begin with a pair of entangled photons BC and DA they are in Phi+ and Psi+ Next we make a bell state measurement on AB. there are four possibles results. According to the result you get, the equality tells you in which Bell State is the CD pair.

Does anybody know how one can say BC are Phi+ and AD in Psi+? I suppose that it can not be known before AB Bell measurement. But how are they post selected?

I agree with naima. The idea of using the labels is to first write the expansion in the "wrong*" order, since that's how it's presented in the book. Then rearrange every term so that the labels are in the order ABCD.

*It's not wrong, since labels instead of ordering are used to indicate ABCD. One can notate either way, which is why I put "wrong" in quotes.

I expanded further and after cancelling I get the following:
|H>(A)|H>(B)|H>(C)|V>(D) + |V>(A)|V>(B)|V>(C)|H>(D) + |H>(A)|V>(B)|H>(C)|H>(D) +
|V>(A)|H>(B)|V>(C)|V>(D)

Should it be that, if the way to verify the equality is to expand as I have done and cancel like terms, and re-arrange the photons as they're originally entangled?