## 5.5 Separable or entangled?

“*Most*” vectors in *cannot* be written as product states

In order to see this, let us write any joint state

Now, for any *product* state, these vectors have a special form.
Indeed, if

Conversely, if ^{107}
So if we want to identify which joint states are product states and which are not, we simply write the joint state according to Equation (*much* smaller than

The problem of deciding whether or not a given state is separable is, in general, a hard problem (i.e. NP-hard).
Because of this, it is interesting to try to understand the notion of separability from different points of view, and it turns out that algebraic geometry yet again has something interesting to say.
The theory relies on the notion of **projective space**, which is a non-trivial topic to try to introduce here, so we do so only briefly, and at a very high speed.

We have repeatedly said that we only really care about state vectors *up to global phase*, i.e. that *the space of lines through the origin*, i.e. of *many* pages to delve into, and so we won’t talk about this point of view here.

Algebraically, it turns out that we can describe the space of such equivalence classes using **homogeneous coordinates**.
Defining **projective n-space** as

*not all simultaneously zero*(i.e. there exists at least one

Why is this useful?
Well, given any pure state *points in the (complex) projective line \mathbb{P}^1 correspond to pure states of a qubit*.

Next, we can always express a pure state of *two* qubits in the form

What is of interest to us here is a particular map known as the **Segre embedding**:
*even more* well behaved than this: as its name suggests, it actually gives an **embedding** (i.e. a “geometric” injection) of the

The image of the Segre embedding is called the **Segre variety**, and you can check that it is given by the set of points
**zero-locus** of a single polynomial).

Now here is the punchline to all this geometric meandering: *a state |\phi\rangle of two qubits is separable if and only if its corresponding point in \mathbb{P}^3 lies in the Segre variety \Sigma*.

For more, see e.g. Cirici, Salvadó, and Taron’s “Characterization of quantum entanglement via a hypercube of Segre embeddings”, arXiv:2008.09583).

Quantum entanglement is one of the most fascinating aspects of quantum theory. We will now explore some of its computational implications.

Even though an entangled state cannot be written as a single tensor product, it can always be written as a

*linear combination*of tensor products, since these form a basis.↩︎