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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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No, the definition of entanglement is not only just that the combined system is not in a product state. This is the case only if the combined system is in a pure state.

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.

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To have true entanglement then do both subsystems have to be mixed? –  user32462 Jan 30 at 10:54
    
If the total system of the two parties is in a pure state, then the two subsystems are entangled if and only if the restricted states of the subsystems are mixed. However, if the total system is already in a mixed state, then also the subsystems can be in a mixed state without being entangled, e.g., they can be in a mixed product state. So to answer your question: to have entanglement, the restricted states of the subsystems must be in a mixed state (but this is an "if and only if" condition when the total system is in a pure state). –  Zoltan Zimboras Jan 30 at 15:26

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