check if the reduced density matrices are pure. This condition is equivalent to the fact that a bipartite pure state has a single Schmidt coefficient. The determination of separability for mixed states is much harder, and currently lacks a complete answer, even in composite systems of dimension as low as ${\mathbb{C}}^2 \otimes {\mathbb{C}}^4$.

To reach a formal definition of separable and entangled states, consider the following preparation procedure of a bipartite quantum state between Alice and Bob. Suppose that Alice prepares her physical system in the state $|e_{i}\rangle$ and Bob prepares his physical system in the state $|f_{i}\rangle$. Then, the combined state of their joint physical system is given by: $$|e_{i}\rangle \langle e_i |\otimes |f_{i}\rangle \langle f_i|.$$ We now assume that they can communicate over a classical channel (a phone line, for example). Then, whenever Alice prepares the state $|e_{i}\rangle$ ($i = 1,2, \ldots, K$), which she does with probability $p_i$, she communicates that to Bob, and correspondingly Bob prepares his system in the state $|f_{i}\rangle$ ($i = 1,2, \ldots, K$). Of course, $\sum_i p_i =1$ and $p_i \geq 0, \ \forall i$. The state that they prepare is then $$\begin{aligned}
\rho_{AB} = \sum_{i=1}^K  p_i |e_i\rangle \langle e_i| \otimes |f_i\rangle 