# What is the general form for GHZ and W class states?

I am reading the following paper on mixed three-qubit states. It states that any three–qubit pure state can be written as (equation 1 of the paper)

$$|\psi_{GHZ}\rangle=\lambda_0 |000\rangle+\lambda_1 e^{i\theta} |100\rangle+\lambda_2 |101\rangle+\lambda_3 |110\rangle+\lambda_4 |111\rangle$$

and W class states have the form (equation 2 of the paper)

$$|\psi_{W}\rangle=\lambda_0 |000\rangle+\lambda_1 |100\rangle+\lambda_2 |101\rangle+\lambda_3 |110\rangle$$

However, later they use the following form for W states (page 2 of the paper)

$$|\psi_{W}\rangle=\kappa_0 |000\rangle+\kappa_1\left(|100\rangle+ |010\rangle+ |001\rangle\right)$$

I did not understand how equation 1 is obtained; also which one amongst equation 2 and the last equation is the "correct" representation of W-class states (or whether these two forms are equivalent to each other). Any help would be greatly appreciated.

The GHZ form of the state follows from a bit of a manipulation after define what you mean by a GHZ state, remembering that all these parametrisations are "up to local unitaries". For me, the GHZ class of states is one that takes the form $$\alpha|u_1\rangle|u_2\rangle|u_3\rangle+\beta|v_1\rangle|v_2\rangle|v_3\rangle,$$ so the first challenge is to see how we can reduce a state of this form, under the action of local unitaries, to the claimed standard form. Pick a convention for the global phase such that the $$|000\rangle$$ term is real. Start by looking at the two basis states $$|111\rangle$$ and $$|011\rangle$$. There must exist a rotation on the first qubit that can completely get rid of the $$|011\rangle$$ term, leaving the $$|111\rangle$$ term real. Now, we can apply the same trick using a single qubit rotation on the second qubit to cancel the $$|010\rangle$$ term out of the pair $$|010\rangle$$ and $$\|000\rangle$$. Finally, use the same trick to cancel the $$|001\rangle$$ term out of $$|001\rangle$$ and $$|000\rangle$$. You'll find the only basis states we have left are those from your statement of $$|\psi_\text{GHZ}\rangle$$, and the only one we've not had the freedom to make real has been the $$|100\rangle$$ term.