quantum mechanics - Definition of Entanglement

The definition of quantum entanglement, found on the internet and the literature is:

On a bipartite system $\mathcal{H}_A \otimes \mathcal{H}_B$, let $\rho$ be a mixed state. It is said to be separable if it is a convex combination of product states

$$\rho = \sum_i \lambda_i \rho^A_i \otimes \rho^B_i$$ Here, $\lambda_i\ge0$, and $\rho^A_i,\rho^B_i\ge0$.

If this is not the case, it is said to be entangled.

My question is, how did they come to this definition? Where did it come from and why does it work? Is there any way to start from physical principles and arive to this definition?