# How do I arrange these two-qubit states based on their entanglement?

How can I arrange the following states in decreasing order based on their entanglement?

i) $$\rho_1 = \frac{1}{2}|0\rangle \langle0|\otimes |+\rangle \langle+| +\frac{1}{2}|+\rangle \langle+|\otimes |1\rangle \langle1|$$

ii) $$\rho_2 = |\psi\rangle \langle\psi|$$, where $$|\psi\rangle=\sqrt{0.99}|00\rangle+0.1|11\rangle$$.

In decreasing order: $$\rho_2$$ then $$\rho_1$$, because no state is less entangled than a separable state like $$\rho_1$$. We know that $$\rho_1$$ is separable, because it is a convex combination of product states.
On the other hand, $$\rho_2$$ is entangled because its Schmidt rank is $$2$$. Moreover, its density matrix
$$\rho_2 = \begin{pmatrix} 0.99 & & & 0.1\sqrt{0.99} \\ & & & \\ & & & \\ 0.1\sqrt{0.99} & & & 0.01 \end{pmatrix}$$
$$\begin{pmatrix} 0.99 & & & \\ & & 0.1\sqrt{0.99} & \\ & 0.1\sqrt{0.99}& & \\ & & & 0.01 \end{pmatrix} \begin{pmatrix}0 \\ 1 \\ -1 \\ 0 \end{pmatrix} = -0.1\sqrt{0.99}\begin{pmatrix}0 \\ 1 \\ -1 \\ 0 \end{pmatrix}.$$