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Let's say I have an arbitrary n-qubit quantum state $|\psi\rangle$. Is there a simple way to compute the states of individual qubits (or sets of qubits if they are entangled) that compose $|\psi\rangle$? E.g. if $n=3$ how can I factorize $|\psi\rangle$ into $|\psi\rangle = |q_0\rangle\otimes |q_1\rangle \otimes |q_2\rangle$ or if let's say $|q_0\rangle$ and $|q_1\rangle$ are entangled $|\psi\rangle=|q_0q_1\rangle\otimes|q_2\rangle$? Or as the case may be, how can I verify it's not factorizable? It's an easy task for the case of 2 qubits, but in general case it looks like I need to test all possible combinations of entangled subsets?

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What you ask is even harder than asking whether a state is separable. The latter is considered a complex problem, sometimes NP-hard. See, e.g., https://en.wikipedia.org/wiki/Separable_state for a quick overview. So, to answer your question, "Is there a simple way...", probably no, given that you assume an arbitrary state $|\psi\rangle$.

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