State Spaces in Classical vs Quantum systems, in the context of classical & quantum computers

Question

I am currently reading "Quantum Computing: A Gentle Introduction" by Rieffel & Polak. In describing the difference between classical and quantum state spaces, they say:

In classical physics, the possible states of a system of n objects, whose individual states can be described by a vector in a two-dimensional vector space, can be described by vectors in a vector space of 2n dimensions. Classical state spaces combine through the direct sum. However, the combined state space of n quantum systems, each with states modeled by two-dimensional vectors, is much larger. The vector spaces associated with the quantum systems combine through the tensor product, resulting in a vector space of $2^{n}$ dimensions.

In the context of general classical physics and general quantum physics, this makes very good sense to me. An object in classical physics can be fully described by its position and momentum (which is the two-dimensional vector space described above) and the time evolution is governed by Hamilton's equations. If we add more objects, the state space grows via the direct sum of the individual vector spaces. For a two-state quantum system however, when we add more particles the overall Hilbert space grows like the direct product of the individual vector spaces and therefore grows in size like $2^{n}$.

What doesn't make sense to me is how this directly relates to classical analog computers and classical digital computers? Take for instance the analog computer example shown here. I suppose it would be possible to convert the equation which it models from a Newtonian form (i.e. $F=ma$) to a Hamiltonian form and perhaps model Hamiltons equations using two coupled active differentiators. Am I to conclude then that this is what is meant by the state space growing by 2n? I'm not sure this is even correct however because there is a friction term I am neglecting (the shock absorber).

And even more confusing, how does this relate to digital computers? If I have a state with 8 bits, then by definition I have 8 bits of information. It seems then that the state space of a digital computer scales like $n$, rather than the $2n$ mentioned previously.

Answer 1

I can only make sense of the paragraph by assuming it's about two completely unrelated systems. The $n=1$ classical system is a particle that can be described by a vector in $\mathbb R^2$, which I suppose is either a position and momentum (as you guessed) or two position coordinates. It's not a bit, nor the state of any sort of classical computer. The $n=1$ quantum system is a qubit. There is no correspondence between the two, so it doesn't make much sense as an analogy.

The state of the analog computer that you linked could be described by the position of the mass and its first time derivative, which could be seen as a vector in $\mathbb R^2$, but I really don't think that's what Rieffel & Polak had in mind. My impression is that they were thinking that if a single qubit is described by a vector in $\mathbb C^2$ then the classical equivalent of it should also be a vector in a two-dimensional space. But, as you said, it isn't.