# What is the defining property for two quantum states to become entangled?

What is the defining property for two quantum states to become entangled? Is it just that the combined system cannot be in a product state? Why is this?

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For the general definition of entanglement consider two systems with respective Hilbert spaces $H_1$ and $H_2$. The composed system has Hibert space $H_1 \otimes H_2$. The state of the full system is described by a density matrix, i.e. a Hermitian positive operator of trace 1 on $H_1 \otimes H_2$. Let $\rho_{12}$ be such a density matrix, and suppose it can be written as a convex combination
$$\rho_{12}=\sum_k p_k \rho_k^{(1)} \otimes \rho_k^{(2)},$$
where $p_k \ge 0$ and $\sum_k p_k =1$, and where the $\rho_k^{(1)}$'s and $\rho_k^{(2)}$'s are density matrices on $H_1$ and $H_2$, respectively. Then we call the state separable . Otherwise it is called entangled.