In general, a bipartite pure state $ρ$ is entangled if and only if its reduced states are mixed rather than pure.
Is this statement generalized to multipartite states? Is there any resource about that?
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$\begingroup$ Simply: yes. But you have to decide what definition of multipartite entanglement you want. Do you want entanglement between all systems, entanglement between each pair for each way you can divide the system into two partitions? $\endgroup$– Quantum MechanicCommented Mar 4 at 17:40
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$\begingroup$ Consider both cases. Can you help me about that? $\endgroup$– rezaCommented Mar 4 at 19:24
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1 Answer
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Depends on the notion of entanglement you're asking about.
With a state like $|000\rangle+|111\rangle$, you can take any bipartition and you'll find non-pure states, and Schmidt coefficients $(1/2,1/2)$. Equivalently, any reduced state has eigenvalues $(1/2,1/2)$ — but note that this doesn't mean all reduced states are maximally mixed. GHZ-like states are entangled in a "maximally global way" in the sense that they are the only pure $n$-qubit states not characterised by their $n-1$ reduced density matrices, see https://arxiv.org/abs/0707.4428.
With a state like $|100\rangle+|010\rangle+|001\rangle$, the reduced states are still non-pure, but now they'll have eigenvalues $(1/3,2/3)$ — still regardless of which one- or two-qubit subsystem you take. These are $W$ states, and are also "maximally entangled" in some sense: they belong to the same entanglement class as the GHZ state, see e.g. How to prove that there are only $2$ entanglement classes of $3$-qubit states? and paper linked there. See also Table 2 in https://arxiv.org/abs/1612.07747.
On the other hand, if you take a state like $(|00\rangle+|11\rangle)|0\rangle$, the question of its entanglement is somewhat trickier. Clearly, the third qubit is separated from the first two, which are maximally entangled. Still, the overall state is not separable, and you can find subsystems which are not pure. Examples like these prompted the definition of genuine entanglement: a state has genuine multipartite entanglement if it's not separable with respect to any bipartition.
Going further, there are many possible quantifiers that can be considered. See e.g. https://arxiv.org/abs/1612.07747. But if you're only interested in genuine multipartite entanglement (of pure states), then you can state the requirement as: there must be no pure reduced state.