Universal Quantum Birthmark: Ghost of the quantum past

TL;DR

The paper addresses how quantum dynamics retain memory of initial conditions even in classically chaotic regimes, challenging the standard ergodic/thermalization picture. It develops a universal birthmark framework by analyzing Haar-random states and spectral statistics via random-matrix theory, deriving $P^{\textrm{UQB}} \ge 2$ (with $P^{\textrm{UQB}} = 2$ for GUE and $P^{\textrm{UQB}} = 3$ for GOE) and a revival factor $P^{\textrm{RQB}} \ge 1$, and shows this persists irrespective of microscopic dynamics. For the full random-matrix case, the paper derives $\bar{P}_{aa}=2/(N+1)$ (GUE) or $3/(N+2)$ (GOE) and $\bar{P}_{ab}=1/N$, yielding $\bar{P}_{aa}/\bar{P}_{ab}=2N/(N+1)$ or $3N/(N+2)$ via Dirichlet statistics and Schur–Weyl invariants. It further shows symmetry constraints modify the bounds via the accessible subspace dimension $d$, with numerical billiard simulations illustrating the theory. The results provide a refined view of quantum ergodicity, highlighting a universal, symmetry-controlled memory in quantum evolution with implications for understanding ergodicity and scarring in chaotic systems.

Abstract

Quantum dynamics retains a permanent and universal memory of its initial conditions, even in systems whose spectra display fully chaotic, random-matrix behavior. This effect, known as the quantum birthmark, appears as an enhancement of the long-time return probability of any non-stationary state compared to the overlap with a typical ergodic state. In this work, we develop the full theoretical foundation for this universal contribution that depends only on the global symmetry class and accessible Hilbert-space dimension, not on the microscopic dynamics. Our findings reveal that quantum evolution preserves an unavoidable, symmetry-controlled imprint of its origin, a quantum effect calling into question classical expectations of ergodicity and the resulting thermalization scenarios.