How to calculate negativity measurement of quantum state?

[munirah]

Good day,

From my reading according to negativity for tripartite state, it is given as below;
$$N_{ABC}(\rho)=(N_{A-BC}N_{B-AC}N_{C-AB})^{1/3}$$

with

$$N_{I-JK}=-2\Sigma_i\sigma_i(\rho^{TI})$$
where
$$\sigma_i(\rho^{TI})$$
being the negative eigenvalues of
$$\rho^{TI}$$,
the partial transpose of $$
\rho
$$
with respect to subsystem $$I$$.

My problems:

1.I not understand how it works. I find the eigenvalues but the value of eigenvalues is positive.But it mention the eigenvalues are negative. For example. I calculate GHZ state and the eigenvalues are positive. What its mean actually?

2.It is same for $$
N_{A-BC}
$$ with another negativity formula given as below,
$$
N(\rho_A)=(Tr[\sigma_{A}^\dagger\sigma_{A}]^{1/2}-1)/2
$$

Please help me.

Thank you

 

[mark726]

.
Thank you for bringing up this interesting topic. The negativity for tripartite states is a measure of entanglement between three subsystems, A, B, and C. It is defined as the geometric mean of the negativity for each bipartite subsystem (A-BC, B-AC, and C-AB). The negativity for a bipartite subsystem is calculated by taking the negative eigenvalues of the partial transpose of the density matrix with respect to that subsystem. This means that the eigenvalues should be negative, not positive.

In the case of the GHZ state, the eigenvalues should indeed be positive. This is because the GHZ state is a maximally entangled state, meaning that it has a high degree of entanglement between all three subsystems. Therefore, the negativity for the GHZ state is zero, as there are no negative eigenvalues.

As for your second question, the formula for $$N_{A-BC}$$ in the second formula you provided is a different measure of entanglement called the logarithmic negativity. This measure is also based on the negative eigenvalues of the partial transpose, but it is calculated differently. The logarithmic negativity is a more sensitive measure of entanglement than the negativity for tripartite states, as it can detect entanglement even when the negativity is zero.

I hope this helps clarify the concepts for you. Please let me know if you have any further questions.Scientist at [Your Institution]