In the linear algebra section of the Qiskit textbook appears the following claim regarding the Bloch sphere:
The surface of this sphere, along with the inner product between qubit state vectors, is a valid Hilbert space.
It is pretty clear that by scaling any quantum statevector we can easily get a vector that points outside or inside the surface of the sphere.. I.e the surface of the sphere isn't a vector space.
From where is this contradiction coming from?
The surface of a Bloch sphere is not a Hilbert space.
Maybe they meant to write that it's a valid projective Hilbert space (in particular it's isomorphic to $\mathbb{CP}^1$)? It's not a vector space, so it cannot be a Hilbert space (note that a "projective Hilbert space" is, perhaps somewhat confusingly, not a Hilbert space).
glS covered the main points, I just wanted to add an important distinction when thinking about qubits, the Bloch Sphere and Hilbert space,
While all valid quantum states exist within a Hilbert they do not form a Hilbert space.
This is why we see only a few valid operations for quantum states (such as inner product and unitary evolution). In contrast, certain fundamental vector operations such as vector addition and scalar multiplication which are critical in defining a vector space are not valid.
All our operations are valid for the Bloch Sphere which represents the actual state space of our qubits, which is clearly not a Hilbert Space (as it is not even a vector space).
I believe that this key distinction between "Existing in a Hilbert Space" and "Forms a Hilbert Space has caused a great deal of misconceptions and confusion in the community.
