# identify a tensor product by virtue of pure and entangled elements

If I take a tensor product of vector spaces (for simplicity - this could be more general) $V\otimes W$ then of course it is a vector space, but it has additional structure. One way to think about this is that there are two distinguished sets $P$ and $E$ where $P=\{\,\mbox{elements of form}\, v\otimes w\,|\, v\in V,w\in W\}$ and E = the complement of P (so in quantum mechanics these are the pure and entangled states). These sets have certain properties, e.g. they are closed under multiplication by non-zero scalars; each of them generates $V\otimes W$ (I think, at least for char $\neq2$, and assuming $E\neq\emptyset$); if dim($V\otimes W$) is prime then $E=\emptyset$; etc.

My question is whether there is a result along the lines of "Given a vector space $X$ with two subsets $P$, $E$ with properties [to be specified], there exist vector spaces $V_1,...,V_n$ and an isomorphism $X\cong V_1\otimes...\otimes V_n$ of vector spaces, such that under that isomorphism the subsets $P$ and $E$ of $X$ correspond respectively to the pure and entangled states of $V_1\otimes...\otimes V_n$"?

(ps this is my first post so apologies if the question is inappropriate/ mis-placed)

• Your question is as far as possible from being inappropriate: it is extremely interesting and very well formulated. It is a great pleasure to welcome you on this site! – Georges Elencwajg Oct 27 '14 at 12:50
• By the way, is it now usual for physicists to interpret entanglement in such abstract terms? When I taught myself basic quantum physics I was very happy when I realized that entanglement had an algebro-geometric interpretation. I mentioned that to a legendary quantum physicist after a talk of his, but obviously he was not familiar with this abstract point of view. – Georges Elencwajg Oct 28 '14 at 12:13
• Well I'm not a real physicist - I started life as a pure mathematician, and now I'm an occasional tourist in theoretical physics - but I would be surprised if it is at all usual. I do wonder whether physics would benefit from being more abstract - I have sometimes struggled to understand things that would have been quite simple to understand if expressed the way a mathematician would express them (eg the way physicists talk about lie algebra representations in quantum mechanics and quantum field theory). Either their way is more useful for doing physics, or they are stuck in a bad equilibrium! – Boaz Moselle Oct 29 '14 at 22:05

Given two vector spaces $V,W$ the space of decomposable tensors in $V\otimes W$ is the Segre cone.
It is often convenient to projectivize the situation to the Segre embedding $i:\mathbb P(V)\times \mathbb P(W)\hookrightarrow \mathbb P(V\otimes W)$, whose image then corresponds to the pure states.
The question you ask is whether you can recognize abstractly in a projective space $\mathbb P(X)$ the image of $i$.
If the dimension of $X$ is 4, the answer is yes: any smooth quadric surface in $\mathbb P^3$ can be seen as the space of pure states , but I don't know the general answer.