Physical significance of the ket vector

Question

I'm trying to understand/form an intuition for why a ket-vector, and its associated bra-vector, are appropriate tools to represent the spin of a qubit.

Generally speaking, if a system has a state/property, that state/property can be represented by a set of numbers that quantify the magnitude (and potentially direction) of that state/property. The magnitude is based on a scale (that hopefully has historical significance and/or has consensus of a large group of people).

Based on this understanding/logic, my intuition of the ket-vector representing qubit spin is that it in order to quantify spin of a qubit, we need 2 real numbers, which can mathematically be represented as a single complex number or a ket-vector. Thus, the ket-vector is just a mathematical tool that helps makes the math a little easier.

But what do the components of the ket-vector signify? Why do we need 2 numbers to specify a single property? I'm having a hard time coming up with other properties/states that need 2 numbers (except maybe waves which need amplitude and frequency to be fully defined - but then again, the properties of waves are amplitude and frequency which are themselves defined by single numbers).

I've read several articles/questions about the components representing the probability amplitude, and how $\alpha_u.\alpha_u^* = $ probability of finding that the spin of the qubit is aligned along $|u>$

This answer talks about how the probability theory used for Quantum Mechanics is slightly different than what we are generally used to, but does not provide any intuition as to why that's the case, and hand-waves it away by talking about interference (which I'm not yet familiar with).

Answer 1

I have waited for someone else to take it up, but there seem to be no takers. So here it comes.

What do you mean by spin of a qubit? Single electron has spin-1/2 and thus only two possible eigenstates. A photon can have two possible circular polarizations, which map onto spin-up/down well. That's simple enough.

But what about a system of coupled electrons, that could have very many possible spin states? Same goes for complex optical beams, where polarization state can be complicated.

My point here is that simple two-state quantum systems occur in nature, and are relatively simple to understand, but they are by far not the only type of quantum system out there. Your question was about a spin of a qubit. I don't know what your source is, but I am guessing that there quibt is defined to be a two-state quantum system. Such system does not need to have a spin at all, but its Hilbert space will be isomorphic to that of a 1/2-spin electron. So this is where the terminology comes from.

Finally, note that in principle you need two complex numbers to define a state of a two-state quantum system. However, conventionally the states of such systems are normalized, and the global phase of the state can be ignored. Then the state of your two-state system is captured by:

$|\psi\rangle = \cos\frac{\theta}{4} |\uparrow\rangle + \exp\left(i\phi\right)\sin\frac{\theta}{4} |\downarrow\rangle$, for $\phi,\,\theta \in [0,2\pi)$