Lets say we have a closed system with states in a Hilbert space $\mathcal{H}$. Every state can be expressed as a sum of energy eigenstates. In a closed system, like a box of atoms, entropy will increase until the entire system is in thermal equilibrium. However, when we write a state as a sum of energy eigenstates, time evolution merely contributes a phase to each basis state. It doesn't seem to me as though states get any more "disordered" over time, so how can the entropy increase?
Let me phrase my question differently. I recall a professor once writing
$$S = k \log (\dim \mathcal{H})$$ on the board. Certainly, the dimensionality of a Hilbert space does not change under time evolution. So how can entropy increase under time evolution? Is the above equation only correct under certain conditions? There must be a big flaw in my reasoning/memory.
Answer
The total entropy of an isolated system indeed does not change under Schrodinger time evolution. To see this, note that (assuming for simplicity that the Hamiltonian does not depend explicitly on time) the system's density matrix satisfies the Von Neumann equation $\rho(t) = e^{-i H t / \hbar}\, \rho(0)\, e^{i H t / \hbar}$, so $\rho(t)$ and $\rho(0)$ are always unitarily equivalent, and therefore have the same eigenvalue spectra. Therefore any entropy measure that depends only on the density matrix weights (which, practically speaking, is all of them), is constant in time.
But the entanglement entropy between subsystems can indeed increase, because the subsystems are not isolated. So if the system, say, starts in a product state with no spatial entanglement between its subsystems, then generically Schrodinger time-evolution will lead to increasing entanglement between the subsystems, so the local entropy associated with each little piece of the whole system will indeed increase, even as the total entropy remains constant. This fact relies on a very non-classical feature of Von Neumann entropy, which is that the sum of the entropies of the subsystems can be greater than the entropy of the system as a whole. (Indeed, in studying the entanglement of the ground state, we often consider systems where the subsystems have very large entanglement entropy, but the system as a whole is in a pure state and so has zero entropy!)
The subfields of "eigenstate thermalization", "entanglement propagation", and "many-body localization" - which are all under very active research today - study the ways in which the Schrodinger time-evolution of various systems do or do not lead to increasing entanglement entropy of subsystems, even as the entropy of the system as a whole always stays the same.
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