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?
$\begingroup$ related: quantumcomputing.stackexchange.com/q/27857/55 $\endgroup$– glS ♦Nov 12, 2022 at 10:44
$\begingroup$ This is probably implicit in your question since you use $|\psi\rangle$, but I assume you're interested in arbitrary pure states? I ask because determining a separability criteria for an arbitrary state is quite hard, even for the bipartite case (nature.com/articles/s41598-018-19709-z) $\endgroup$– VisipiNov 12, 2022 at 20:46
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 $|\psi\rangle$.