How many $n$-qubit states are entangled?

Question

My question is a bit broad, but my concern is mainly on understand the ratio between the number of possible linear combinations that can be decomposed in a direct product of states and the number of possible entangled states in that system. The system is a just an $n$-qubits set, as an usual vector space, with the assumption that all the states in a certain superposition can be expected as equally probable. As an example to clarify what I mean take:

$$ \left| \psi \right\rangle = \frac{\sqrt{2}}{2} (\left| 00 \right\rangle + \left| 11 \right\rangle) $$

I know that the ratio in a 2-qubits system should be 50%, but what about $n=100$?

To improve the clarity of the question I would ask, how do you interpret this kind of graph?

Where Product = a state that can be decomposed uniquely by a tensor product. Entangled = it's own defition. Is this a true picture of what's going on in an $n$-qubit system or not?

Answer 1

The probability of finding an entangled state while randomly sampling (from the uniform distribution) over pure states is one: almost all pure states are entangled.

This is shown in this answer on physics.SE. This paper might also be of interest.

The gist is that separable states correspond to rank-1 matrices, which have zero measure in the set of all matrices. This is also discussed in this post on math.SE.

Another less rigorous argument could be to observe that separable states are always infinitesimally close to entangled out: given an arbitrary separable pure state $|\Psi\rangle\equiv|\psi\rangle\otimes|\phi\rangle$, take $|u\rangle,|v\rangle$ such that $\langle u|\psi\rangle=\langle v|\phi\rangle=0$. Then, the state $$\frac{1}{\sqrt{1+\epsilon^2}}\left[|\Psi\rangle+\epsilon(|u\rangle\otimes|v\rangle)\right]$$ is entangled for all $\epsilon>0$ and can be made to be infinitesimally close to $|\Psi\rangle$. This hints towards the set of separable states having empty interior (although technically, having empty interior is not in itself enough to prove having zero measure, see e.g. this blog post).