First of all: I apologise in advance for if my question will be arid, wrong written or even nonsensical.


I was at a talking with a professor last week, and the question of "Entanglement and Algebraic Geometry" came out.
What emerged really fascinated me, so I'm here to ask more clarifications, explanations or even references.


Here is what I "understood" and wrote down on a piece of paper (that is, heard and hence reporting here):


"Say we have two systems made of states. It's easy to understand what entanglement is from a geometric point of view. Those systems can the thought as objects in $\mathbb{P}^2(\mathbb{C}^3)$ (projective space). The states they are made of can be pure or not pure. We can see the pure states are basically quadrics in $\mathbb{P}^2(\mathbb{C}^3)$, while the non pure states are what are called the entangled ones".


So I'm asking you: can you explain me better the above statement, assuming it's not meaningless? Or provide me some reference, paper, books...


Thank you so much!



Entanglement, quadrics and $\mathbb{P}^2(\mathbb{C}^3)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$Let me just make a couple of comments on the question to clean up the confusion about the correct dimension of the projective space — it should be $\mathbb{P}^3(\mathbb{C})$. The state space of a qubit is the Hilbert space $\mathbb{C}^2$, and the $n$-qubit is obtained as tensor product of $n$ copies of $\mathbb{C}^2$. There are different choices for symmetry group, one possible is SLOCC (stochastic local operations with classical communication) which for the $n$-qubit would be the group $\SL\_2(\mathbb{C})^{\times n}$ acting on $(\mathbb{C}^2)^{\otimes n}$. For the 2 qubits, this would be $\SL\_2(\mathbb{C})\times\SL\_2(\mathbb{C})$ acting on $\mathbb{C}^2\otimes\mathbb{C}^2$. The action can be identified with the group acting on $2\times 2$-matrices by left and right multiplication. The orbits of the group on the projective space associated to $\mathbb{C}^2\otimes\mathbb{C}^2$ (that is the $\mathbb{P}^3$ that should appear in the question) correspond to the different states the system can be in.


Now algebraic geometry tells us that there are two orbits for the action of $\SL\_2(\mathbb{C})\times\SL\_2(\mathbb{C})$ acting on $\mathbb{P}^3$. One open orbit, isomorphic to $\PSL\_2(\mathbb{C})$, and one closed orbit isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$ — that is the quadric mentioned in Tom Ducat's [comment](https://mathoverflow.net/questions/450048/entanglement-quadrics-and-mathbbp2-mathbbc3#comment1163095_450048). The closed orbit can be obtained as the orbit of an elementary tensor, so it consists of pure states. The open orbit consists of the entangled states, it is the orbit of the so-called Bell state.


There is further literature on this, people have been working on using geometric invariant theory etc to understand polynomial invariants which separate different entangled states in $n$-qubits. My knowledge there is a bit dated, but it seemed that 4 qubits is the state of the art.


