I have a question, which may be a very simple to answer for people who actually studied physics. Unfortunately I didn’t so this question perplexes me. The question is “how does (classical) chaos come about given that everything has to be consistent with quantum mechanics?”.
Chaos is when a very small difference in the initial state of a system can lead to very different states of the system after some amount of time. Suppose we begin with two quantum states 
This is probably a very silly question with a very simple answer, but I’d appreciate the answer nonetheless.
Not simple! People have been arguing about this for ages. I like the work of Salman Habib: http://t8web.lanl.gov/people/salman/ndyn.html
I think you need to extend your analysis a bit here. If you are considering the system as a whole, then it is clear that the system is not chaotic. It’s not clear to me that this should hold when you consider a subsystem.
I agree Joe. But suppose you had a system that classically exhibits chaos and you were able to perfectly isolate it from the environment, would that system exhibit chaos, or does the chaos really emerge because you are tracing out a part of the total system?
How about this isolated quantum system: a quantum computer which is simulating a classical computer which in turn is running some computer code which simulates a chaotic system? Both the original classical computer and the quantum computer that simulates it are producing the same apparently-chaotic behaviour.
I find quite intriguing to look at this from the (algorithmic) entropic point of view. The quantum evolution is unitary, so once you know the initial state, the complexity of a time evolved state can not grow faster than a logarithmic function of the elapsed time, that is no possible chaos.
Once you add an environment, it “measures” the system, rolling dices each time and making the state of the system more random, so that you recover chaotic dynamics.
You’re right, quantum chaos is different from classical in many aspects. But it exists in the sense that the dynamics of the quantum regime shows different features depending on the existence of chaos in the corresponding classical regime. The way you stated chaos, as a divergence in distance between two neighboring points in phase space evolving with the same dynamics, can also be stated in another way: comparing the trajectories of only one point in phase space with two slightly different hamiltonians. In this case, the quantum state overlap as you wrote depends on the perturbation, and one finds different results for regular or chaotic dynamics. This is a kind of echo experiment, also referred to as fidelity decay. What happens is that the overlap of the differently evolved states decays faster for regular dynamics than for the chaotic one. Other interesting things that happen in quantum chaos is Anderson localization, when the wavefunction localizes instead of spreading like its classical counterpart, and chaos assisted tunneling, when the state tunnels between different stable regions of phase space if there is chaos in other regions around.