## 8.2 Pauli stabilisers

The stabiliser (or stabilizer, if you like) formalism is an elegant technique that is often used to describe vectors and subspaces.
Suppose you want to specify a particular vector in a Hilbert space.
The most conventional way to do this would be to pick a basis and then list the coordinate components of the vector.
But we could instead list a set of operators that leave this vector invariant.
More generally, we can define a vector subspace (rather than just a single vector, which corresponds to a 1-dimensional subspace: its span) by giving a list of operators that fix this subspace.
Such operators are called **stabilisers**.

We say that an operator **stabilises** a (non-zero) state **stabiliser state** (or **stabiliser subspace**).

In other words, an operator *have* to pay attention to the global phase factor: if *not* say that

For example, we can look at states stabilised by the Pauli operators with factors

On the Bloch sphere, these single-qubit stabiliser states lie at the intersection of the three axes with the surface of the sphere.

We can also say something about the remaining two elements of the single-qubit Pauli group:

The set of all stabilisers of a given state or given subspace form a group: if **stabiliser group** of the given state or subspace.

Using this language, we can rephrase the previous example by saying that the stabiliser group of the state

For our purposes, we are only really interested in stabilisers that are also elements of the *abelian* group.
It turns out that such stabilisers can describe highly entangled states.
In particular, the four Bell states (which we first talked about in Section 5.6.1) can be defined rather succinctly by their stabiliser groups:

Bell state | Stabiliser group |
---|---|

Not only this, but some vector spaces are also rather easily defined: the subspace of the three-qubit state space spanned by

Right now, it might seem more complicated to use stabilisers to define vectors or subspaces, but when we start looking at states with a larger and larger number of components we will see how this approach ends up being very tidy indeed!
It is not be true that the stabiliser description of states and subspaces will *always* be the most concise, but it is true in a lot of cases that are of interest to us.

Returning to our claim that stabiliser groups that are subgroups of

An ** n-qubit Pauli stabiliser group** is any subgroup of

**Pauli stabilisers**.

Recall that, in order for the subspace *commute*.
Indeed, if *some* subspace

The size of any Pauli stabiliser ^{155}
**presentation**.
In order to choose a presentation from the set of elements of *ordered* sets (i.e. we are keeping track of the order in which we pick the elements, so

For example, the Bell state

So now we know the size of a Pauli stabiliser, but what can we say about the dimension of the subspace that it stabilises?
If

There is a more geometric way of understanding why powers of 2 keep on turning up in these calculations.
Given independent Pauli generators, it is convenient to think about the state or subspace that they stabilise as being the result of repeatedly bisecting the Hilbert space.
Let

An interesting small exercise here is to explain why the product of any

*independent*Pauli stabilisers cannot be equal to the identity.↩︎