Changing Basis States

Before we get too far off in the QIS direction, let’s tackle the fundamental issue of representing quantum states using different sets of basis vectors. In fact, a superposition state in one basis set will be a single basis state in some other appropriately chosen basis set. More importantly, as we have mentioned, we get more information from a quantum state if we make measurements on it relative to several basis sets. For example, we might start with the description of the polarization state for photons using horizontal and vertical polarization basis states but want to make measurements in terms of $+45\degree$ and $−45\degree$ polarization orientations. So, how do we express a qubit state in terms of these new basis states?

Let’s warm up by considering ordinary two-dimensional vectors, like force vectors in physics. The actual physical interpretation is not important, but it may be helpful to have a concrete example in front of us.

Let’s look at the figure below, where we can see the $x$ and $y$ components of the vector $\vec{F}$ .