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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.

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.

Let the Werner state $$\rho_W = W\mid\Psi^-\rangle\langle\Psi^-\mid + \frac{1-W}{4}\mathbb{I},\ W\in [0,1].$$ 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].

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.

Let the Werner state $\rho_W = W\mid\Psi^-\rangle\langle\Psi^-\mid + \frac{1-W}{4}\mathbb{I},\ W\in [0,1]$. 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/pdf/1303.3081.pdf].

Let the Werner state $$\rho_W = W\mid\Psi^-\rangle\langle\Psi^-\mid + \frac{1-W}{4}\mathbb{I},\ W\in [0,1].$$ 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].