Rearranging equation in Dirac-notation for 3 particles (quantumteleportation)

[keen23]

Hello all!
I try to follow the computation in my textbook (nielsen, quantum computation) and miss a step.

Homework Statement

They say the following state
$$|p\rangle=\frac{1}{2}\big[a (|0\rangle+|1\rangle)(|00\rangle+|11\rangle)+b(|0\rangle-|1\rangle)(|10\rangle+|01\rangle)\big]$$
could be rearranged to
$$|p\rangle=\frac{1}{2}\big[|00\rangle(a|0\rangle+b |1\rangle)+|01\rangle(a|1\rangle+b|0\rangle)+|10\rangle(a|0\rangle-b|1\rangle)+|11\rangle(a|1\rangle-b|0\rangle)}\big]$$.

But I don't see how.

(The start state comes from combining an arbitrary state a|0>+b|1> with an epr-pair in bell-state, then using CNOT for particles 1 and 2, then Hadamardgate on particle 1, well, I think that's not important for my question).

Homework Equations

The Attempt at a Solution

With normal expansion I get:
$$|p\rangle=\frac{1}{2}\big[a|0\rangle|00\rangle+a|0\rangle|11\rangle+a|1\rangle|00\rangle+a|1\rangle|11\rangle+b|0\rangle|10\rangle+b|0\rangle|01\rangle-b|1\rangle|10\rangle-b|1\rangle|01\rangle$$
$$=|00\rangle a(|0\rangle +|1\rangle )+|01\rangle b(|0\rangle -|1\rangle )+|10\rangle b(|0\rangle -|1\rangle )+|11\rangle a(|0\rangle +|1\rangle )$$

So "normal" expansion seems not to be the right way, but how shall I do it? Even if I think about the physical meaning I don't see what's wrong.
Maybe someone has more experience?
Thanks for your help!

 

[Hurkyl]

I think your mistake is that you're setting |00>|1> and |1>|00> equal -- but surely the first one equals |001> and the second equals |100>, which are different?

 

[keen23]

Oh no! Yes, sure. that was my mistake.
Thank you!