My text introduces multi-quibt quantum states with the example of a state that can be "factored" into two (non-entangled) substates. It then goes on to suggest that it should be obvious1 that the joint state of two (non-entangled) substates should be the tensor product of the substates: that is, for example, that given a first qubit
$$\left|a\right\rangle = \alpha_1\left|0\right\rangle+\alpha_2\left|1\right\rangle$$
and a second qubit
$$\left|b\right\rangle = \beta_1\left|0\right\rangle+\beta_2\left|1\right\rangle$$
any non-entangled joint two-qubit state of $\left|a\right\rangle$ and $\left|b\right\rangle$ will be
$$\left|a\right\rangle\otimes\left|b\right\rangle = \alpha_1 \beta_1\left|00\right\rangle+\alpha_1\beta_2\left|01\right\rangle+\alpha_2\beta_1\left|10\right\rangle+\alpha_2\beta_2\left|11\right\rangle$$
but it isn't clear to me why this should be the case.
It seems to me there is some implicit understanding or interpretation of the coefficients $\alpha_i$ and $\beta_i$ that is used to arrive at this conclusion. It's clear enough why this should be true in a classical case, where the coefficients represent (where normalized, relative) abundance, so that the result follows from simple combinatorics. But what accounts for the assertion that this is true for a quantum system, in which (at least in my text, up to this point) coefficients only have this correspondence by analogy (and a perplexing analogy at that, since they can be complex and negative)?
Should it be obvious that independent quantum states are composed by taking the tensor product, or is some additional observation or definition (e.g. of the nature of the coefficients of quantum states) required?
1: See (bottom of p. 18) "so the state of the two qubits must be the product" (emphasis added).
Best Answer
Great question! I don't think there is anything obvious at play here.
In quantum mechanics, we assume that that state of any system is a normalized element of a Hilbert space $\mathcal H$. I'm going to limit the discussion to systems characterized by finite-dimensional Hilbert spaces for conceptual and mathematical simplicity.
Each observable quantity of the system is represented by a self-adjoint operator $\Omega$ whose eigenvalues $\omega_i$ are the values that one can obtain after performing a measurement of that observable. If a system is in the state $|\psi\rangle$, then when one performs a measurement on the system, the state of the system collapses to one of the eigenvectors $|\omega_i\rangle$ with probability $|\langle \omega_i|\psi\rangle|^2$.
The spectral theorem guarantees that the eigenvectors of each observable form an orthonormal basis for the Hilbert space, so each state $\psi$ can be written as $$ |\psi\rangle = \sum_{i}\alpha_i|\omega_i\rangle $$ for some complex numbers $\alpha_i$ such that $\sum_i|\alpha_i|^2 = 1$. From the measurement rule above, it follows that the $|\alpha_i|^2$ represents the probability that upon measurement of the observable $\Omega$, the system will collapse to the state $|\omega_i\rangle$ after the measurement. Therefore, the numbers $\alpha_i$, although complex, do in this sense represent "relative abundance" as you put it. To make this interpretation sharp, you could think of a state $|\psi\rangle$ as an ensemble of $N$ identically prepared systems with the number of $N_i$ elements in the ensemble corresponding to the state $|\omega_i\rangle$ equaling $|\alpha_i|^2 N$.
Now suppose that we have two quantum systems on Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$ with observables $\Omega_1$ and $\Omega_2$ respectively. Then if we make a measurement on the combined system of both observables, then system 1 will collapse to some $|\omega_{1i}\rangle$ and system 2 will collapse to some state $|\omega_{2 j}\rangle$. It seems reasonable then to expect that the state of the combined system after the measurement could be any such pair. Moreover, the quantum superposition principle tells us that any complex linear combination of such pair states should also be a physically allowed state of the system. These considerations naturally lead us to use the tensor product $\mathcal H_1\otimes\mathcal H_2$ to describe composite system because it is the formalization of the idea that the combined Hilbert space should consists of all linear combinations of pairs of states in the constituent subsystems.
Is that the sort of motivation for using tensor products that you were looking for?