## Where does chaos come from?

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 $|psi_1rangle$ and $|psi_2rangle$. The ‘distance’ between these two initial states is simply $|langle psi_1|psi_2rangle|^2$. If we evolve each of these initial states under some arbitrary unitary evolution, then the distance between the output states is $|langle psi_1|U^dag U|psi_2rangle|^2 = |langle psi_1|psi_2rangle|^2$. That is, unitary evolution simply preserves the distance between the two states. So given this constraint, that unitary evolution always preserves the distance between states, how is it that we can observe chaos is the classical world.

This is probably a very silly question with a very simple answer, but I’d appreciate the answer nonetheless.