# How to factorize a multi-qubit quantum state?

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?

• – glS
Commented Nov 12, 2022 at 10:44
• 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) Commented Nov 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 $$|\psi\rangle$$.