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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.

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  • $\begingroup$ not the usual definition or notation of the Werner states: en.m.wikipedia.org/wiki/Werner_state $\endgroup$ Jun 25, 2019 at 0:22
  • $\begingroup$ what is your $|\Psi^{-}\rangle$ ? $\endgroup$
    – wcc
    Jun 25, 2019 at 1:41
  • $\begingroup$ $\frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$ $\endgroup$
    – SescoMath
    Jun 25, 2019 at 2:21
  • $\begingroup$ @PhysMath Such information should be added to the post, not just left in a comment! $\endgroup$ Jun 25, 2019 at 11:14
  • $\begingroup$ @ZeroTheHero That's a perfectly valid Werner state for qubits. $\endgroup$ Jun 25, 2019 at 11:14

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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 $W>\frac{1}{3}$.

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