Entanglement of Werner States

Question

Let the Werner state $$\rho_W = W\mid\Psi^-\rangle\langle\Psi^-\mid + \frac{1-W}{4}\mathbb{I},\ W\in [0,1],$$ where $|\Psi^-\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$. I have repeatedly heard that such a state is separable if $W\leq\frac{1}{3}$ and entangled otherwise. I would like to see where exactly the $\frac{1}{3}$ matters. I know the definition of entanglement, but am having trouble working with the density matrix form. Could someone show me where to begin?

This form of the Werner state is used in https://arxiv.org/abs/1303.3081.

Answer 1

Self-Answer

Using the positive partial transposition criterion, we claim that $\rho_W$ is entangled if the smallest eigenvalue of its partial transpose is positive.

$\begin{align*}\rho_W &= \frac{1}{2}\begin{pmatrix} 0&0&0&0\\ 0&W&-W&0\\ 0&-W&W&0\\ 0&0&0&0 \end{pmatrix} + \frac{1}{4}\begin{pmatrix} 1-W&0&0&0\\ 0&1-W&0&0\\ 0&&1-W&0\\ 0&0&0&1-W \end{pmatrix}\\ &= \frac{1}{4}\begin{pmatrix} 1-W&0&0&0\\ 0&W+1&-2W&0\\ 0&-2W&W+1&0\\ 0&0&0&1-W \end{pmatrix}. \end{align*}$

The partial transpose is $\rho_W^{T_B} = \frac{1}{4}\begin{pmatrix} 1-W&0&0&-2W\\ 0&W+1&0&0\\ 0&0&W+1&0\\ -2W&0&0&1-W \end{pmatrix}$.

It has eigenvalues $\frac{1-3W}{4}$ (with multiplicity 1) and $\frac{W+1}{4}$ (with multiplicity 3).

Thus, the state is entangled when $W>\frac{1}{3}$.