What does it mean to span the Bloch sphere?

[PieroC]

If I construct a set of qubit gates, say {G₁, G₂ ... G_k ... G_n}, that can act on a state |ψ>, what does it mean for the set of states G_k |ψ> to span the Bloch sphere?

As an example, take the set {G₁, G₂, G₃, G₄} = { I, X _π/2 , Y _π/2, X_π }

Here, X _π/2 denotes a π/2 rotation about the x-axis, Y _π/2 denotes a π/2 rotation about the y-axis, and so on.
The set of states |ψ_k>= G_k |ψ>, is said to span the Bloch sphere. But I'm having trouble understanding what this really means.

 

[PeterDonis]

what does it mean for the set of states Gk |ψ> to span the Bloch sphere?

It means that every state in the Bloch sphere is in the set of states $G_k \vert \psi \rangle$.

 

[Strilanc]

Where did you read that those specific operations span the Bloch sphere? To me it looks like they don't, because the set of rotations you can do by combining 90 degree turns is finite and pretty small. There will be plenty of states on the Bloch sphere that you can't get arbitrarily close to.

 

[PieroC]

Where did you read that those specific operations span the Bloch sphere?

This article on quantum gate set tomography https://arxiv.org/pdf/1509.02921.pdf mentions example gate sets on pg 20. The gate set I described above is the first example used. To quote the article, "it is easy to see that the set of states F_k |ρ>> spans the Bloch sphere for any pure state |ρ>>."

To me it looks like they don't

I am also not convinced.

 

[Strilanc]

Oh, in that paper it looks like they're using "span" to mean "we can characterize any state on the Bloch sphere by measuring the computational-basis expectation after applying each of these operators" or something similar to that.