I have been working on a project during which the opportunity arose to work with 2-dimensional vectors as 3-dimensional qubits. The problem I soon encountered was that all the vectors had lengths shorter than one. It’s been challenging to find information about how to work with this situation, which is why I am sharing what I have learned about it thus far. I am hoping that social media threads might open up insights into working in even higher dimensions, which is something I am now actively working on.
Inside the Bloch Sphere
A Bloch Sphere, the 3-dimensional graphical representation of a qubit, is a unit sphere. A unit sphere is a sphere with a radius of one, which simplifies certain calculations. If we cut the Bloch Sphere in half at the equator, we have a surface formed by a 2-dimensional x-y plane. The x-axis runs from the + state to the - state and the y-axis runs from the i state to the -i state.
Now imagine having a set of 2-dimensional points on this x-y plane, as shown below. “Quantity of item 1" is the x-axis, and “quantity of item 2" is the y-axis. The actual datapoints don’t matter; the important visualization is that the gray points are well inside the surface of the sphere.
