# How to differentiate GHZ state and W state?

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1. Oct 24, 2016

### munirah

• OP warned about not using the homework template
Good day,

I try to differentiate GHZ-state and W-state using three tangle. Suppose The value three tangle for GHZ-state equal to 1 while W-state equal to 0.

I used three tangle formula,

$$\tau_{ABC}=\tau_{A(BC)}-\tau_{AB}-\tau_{AC}=2(\lambda^{AB}.\lambda^{AB}+\lambda^{AC}.\lambda^{AC})$$

where the the

$$\lambda^{AB}.\lambda^{AB}$$ and $$\lambda^{AC}.\lambda^{AC}$$ are eigenvalues of

$$\rho_{AB}.\rho^{\sim}_{AB}=(\sigma_y\otimes\sigma_y\rho^*_{AB}\sigma_y\otimes\sigma_y)$$

where the asterisk denotes complex conjugation in the standard basis and $$\sigma_y=\begin{pmatrix} 0 &-i & \\i & 0 \end{pmatrix}$$

I measure the W-state and the density matrix of W-state:

$$\rho_w=\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{1}3 & \frac{1}3& 0 & \frac{1}3 & 0 & 0 & 0 \\0 & \frac{1}3 & \frac{1}3& 0 & \frac{1}3 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{1}3 & \frac{1}3& 0 & \frac{1}3 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$$

and the

$$\rho_{AB}=\rho_{AC}= \begin{pmatrix} \frac{1}3 & 0 & 0 & 0 \\0 & \frac{1}3 & \frac{1}3 & 0 \\0 & \frac{1}3 & \frac{1}3 & 0\\0 & 0& 0 & 0 \end{pmatrix}$$

Since it real matrix I assume it a multiplication matrix between $$\rho_{AB}.\rho^{\sim}_{AB}$$.

I get the eigen values= 2/3 and 1/3

and the value of three tangle for W-state = 0.8888... not 0

Where the wrong I did? I'm really stuck. Please help me to show the way to calculate three tangle correctly.

Thank you

2. Oct 29, 2016