Dynamical quantum ergodicity from energy level statistics

Abstract

Ergodic theory provides a rigorous mathematical description of classical dynamical systems and in particular includes a formal definition of the ergodic hierarchy consisting of merely ergodic, weakly-, strongly-, and K-mixing systems. Closely related to this hierarchy is a lessknown notion of cyclic approximate periodic transformations [see, e.g., I. Cornfield, S. Fomin, and Y. Sinai, Ergodic theory (Springer-Verlag New York, 1982)], which maps any "ergodic" dynamical system to a cyclic permutation on a circle and arguably represents the most elementary notion of ergodicity. This paper shows that cyclic ergodicity generalizes to quantum dynamical systems, and this generalization is proposed here as the basic rigorous definition of quantum ergodicity. It implies the ability to construct an orthonormal basis, where quantum dynamics transports an initial basis vector to all other basis vectors one by one, while minimizing the error in the overlap between the time-evolved initial state and a given basis state with a certain precision. It is proven that the basis, optimizing the error over all cyclic permutations, is obtained via the discrete Fourier transform of the energy eigenstates (for Hamiltonian systems) and quasienergy eigenstates (for Floquet systems). This relates quantum cyclic ergodicity to level statistics. We then show that Wigner-Dyson level statistics implies quantum cyclic ergodicity, but that the reverse is not necessarily true. For the former, we study CUE, COE, and CSE ensembles and quantitatively demonstrate the relation between spectral rigidity and ergodicity. For the latter, we study an irrational flow on a 2D torus and argue that both classical and quantum flows are cyclic ergodic. However, the corresponding level statistics is neither Wigner-Dyson nor Poisson. Finally, we use the cyclic construction to motivate a quantum ergodic hierarchy of operators and argue that under the additional assumption of Poincaré recurrences, cyclic ergodicity is a necessary condition for such operators to satisfy the eigenstate thermalization hypothesis. This work provides a general framework for transplanting some rigorous results of ergodic theory to quantum dynamical systems.

Information scrambling and the growth of errors in noisy tomography - a quantum signature of chaos

“…Therefore, while classically chaotic dynamics generate classical information in the form of classical trajectories, quantum chaotic dynamics generate quantum information in the form of pseudo-random vectors in the Hilbert space, which typically have a high entropy. Vigorous thrust in the understanding of quantum many-body dynamical systems * madhok@physics.iitm.ac.in through dynamically generated entanglement [4][5][6][7][8][9][10] and quantum correlations [11,12], deeper studies in the ergodic hierarchy of quantum dynamical systems [13][14][15][16] have been some recent milestones. Also, the out-of-timeordered correlators (OTOCs) that attempt to capture operator growth and scrambling of quantum information have been very useful as a probe for chaos in quantum systems [17][18][19][20][21][22][23][24].…”