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
$$ \rho_2 = \begin{pmatrix} 0.99 & & & 0.1\sqrt{0.99} \\ & & & \\ & & & \\ 0.1\sqrt{0.99} & & & 0.01 \end{pmatrix} $$
fails Peres-Horodecki criterion, because
$$ \begin{pmatrix} 0.99 & & & \\ & & 0.1\sqrt{0.99} & \\ & 0.1\sqrt{0.99}& & \\ & & & 0.01 \end{pmatrix} \begin{pmatrix}0 \\ 1 \\ -1 \\ 0 \end{pmatrix} = -0.1\sqrt{0.99}\begin{pmatrix}0 \\ 1 \\ -1 \\ 0 \end{pmatrix}. $$
