What is the logarithmic Negativity of the Werner state?

Question

What is the Logarithmic Negativity of the Werner state

$$\rho_w = p|\Psi^-\rangle\langle\Psi^-|+\frac{1-p}{4}I_4$$

where $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|10\rangle-|01\rangle)$?

Answer 1

Hint 1: Logarithmic negativity is easily computed once we know the eigenvalues of $\rho_w^\Gamma$ - where $^\Gamma$ denotes the partial transpose - since then it becomes a matter of substitution into the formulas in the Wikipedia article.

Hint 2: Finding the eigenvalues becomes easier once we observe that if $|v\rangle$ is an eigenvector of $\rho_w^\Gamma$ then $|v\rangle$ is an eigenvector of $(|\Psi^-\rangle\langle\Psi^-|)^\Gamma$.