# What is a general entangled bipartite system?

Is there a general pure entangled bipartite system that all other entangled bipartite systems are a special case of it?

• It's much easier the other way: there is a general separable bipartite state $\sum_i p_i \hat{\rho}_i\otimes\hat{\sigma}_i$ such that all other separable bipartite states are a special case of it. Then entangled states are all of the states not belonging to this. Commented Sep 7, 2023 at 13:56
• I want a same formula for entanglement, if it is exist.
– reza
Commented Sep 7, 2023 at 14:07
• What does a state being a "special case" of another state mean? Commented Sep 7, 2023 at 20:17
• It means u can make any pure entangled bipartite state via it.
– reza
Commented Sep 7, 2023 at 20:21

I didn't see the question asked about pure states - then yes! A general entangled pure state takes the following form, via the Schmidt decomposition: $$|\Psi\rangle=\sum_i \psi_i |a_i\rangle\otimes |b_i\rangle$$ for some non-negative coefficients $$\psi_i$$ and some pure states $$|a_i\rangle$$ and $$|b_i\rangle$$. To be entangled, more than one of the coefficients $$\psi_i$$ must be nonzero. This is a bit of a strange way of writing the set, but we can be clever and say that any state of the form of $$|\Psi\rangle$$ with all Schmidt coefficients $$|\psi_i|<1$$ is entangled. Unentangled states are those with the same form but with one of the coefficients having magnitude equal to unity.