# Why a single qubit is a state in 2D Hilbert space and not 3D or higher?

A physical qubit, for example, an electron with energy level (ground state and excited state) represents a simple quantum system. I was curious whether a physical system is a basis for mathematically representing qubit in 2D space?

Why not a qubit state be in 3D Hilbert space? Is there a mathematical formulation for this even though an equivalent quantum system doesn't exist?

Note: I don't have a formal background in physics or mathematics, this question came to mind when I started reading about quantum computation.

• You'd probably find photon polarization more intuitive than electron spin (I recommend watching that video). There's only two basis states for an electron (up and down spin states), so it's natural that they live in a 2D Hilbert space, rather than a 3D or higher space. The dimension of a vector space (complex Hilbert space, in this context) is determined solely from the number of basis states. Dec 14 '19 at 15:19
• @SanchayanDutta Thanks for the link. I will watch. Still, based on response, I feel question is not conveyed clearly or I missed the point in the answer. I mentioned "electron example" just to bring physical and mathematical world. If we ignore physical realization, then extending dimension makes sense (just as a theory)? Dec 14 '19 at 15:50
• In theory, you could consider a higher dimensional complex Hilbert space in which the 2-dimensional Hilbert space of a qubit is embedded. But I don't see the use. We generally consider the smallest vector space structure in which the physical description is meaningful. Dec 14 '19 at 15:56
• Ground state and excited state are two states which span 2D Hilbert space; if you have 3 independent states (say ground state, first excited state and second excited state) then you get 3D Hilbert space. Dec 14 '19 at 16:11
• Thanks for the clarification. How to use such a formulation would be interesting I guess! Dec 14 '19 at 16:26

• @KittuA For example, if you want to describe the density matrix of a single qubit, that is a Hermitian, non-negative matrix with trace 1. (It's the trace that ensures the probabilities sum to 1). So you can write it as $\left(\begin{array}{cc} z & x+iy \\ x-iy & 1-z\end{array}\right)$i.e. just 3 real parameters $x,y,z$. Nov 29 at 10:29