Quantum eigenstates from classical Gibbs distributions

We thank the referees for providing valuable feedback. Following their suggestions we rearranged the material to highlight the new results related to the classical Gibbs ensemble. We rearranged the material such that original derivations and the main results of the paper appear sooner, moving some of the introductory material and the results on microcanonical ensembles to the Appendices. Next to some minor textual changes, we also added some additional topics suggested by the referees and also by our other colleagues, in particular by Sir. Michael Berry. The key changes can be found below.

- We have rearranged the structure of the paper such that the new results on the Gibbs distribution appear earlier. Introductory sections on the dynamics and microcanonical ensemble have been moved to Appendix.

- We have added a derivation of the two leading-order corrections in $\beta$ to the Gibbs Hamiltonian (Sec. 4.2). These corrections highlight that this is {\emph not} a semiclassical expansion: different orders in $\beta$ have the same order in $\epsilon$. We tested these corrections in the example of tunneling and even for $\beta=1$, i.e. when the small parameter is not that small, we observed that the relative mistake in the tunneling gap can be reduced by orders of magnitude by taking into account higher-order corrections (see the inset in Fig. 6).

- We have added a derivation of the Gibbs Hamiltonian for a particle in the presence of a vector potential and showed how, to leading order in $\beta$, the resulting Gibbs Hamiltonian again agrees with the quantum Hamiltonian. For this reason, all Berry-type phases encoded in the quantum states also appear to be contained in the classical eigenstates, extending the applicability of this formalism. We illustrated the agreement between quantum and classical eigenstates for a confined particle particle in a 2D potential in the presence of a magnetic field and found excellent correspondence both in terms of absolute values and the phases of the wave functions (Fig. 9). Furthermore, we also added an exact analytic derivation of Landau Levels from the classical Gibbs ensemble in the presence of a constant magnetic field (Sec. 5.5.).

- We have included a discussion of chaotic and integrable two-dimensional potentials, which was independently suggested by several people including the third referee. We found that the classical spectrum indeed satisfies the BGS and Berry-Tabor conjectures (Sec. 6). It is remarkable that having the Gibbs distribution alone in hand and without analyzing any dynamics, one can determine whether the classical system is chaotic or not.

- Minor textual changes following the Referees' comments (including corrected typos, an explanation of how all numerical results were obtained, and a short comment on the Koopman-von Neumann construction).