Quantum Adiabatic Theorem under Wiener Noise: Maximal Mean-Square Estimates

Abstract

We derive maximal mean-square adiabatic estimates for finite-dimensional stochastic Schrödinger equations with Hamiltonian-proportional Wiener noise. For deterministic $C^2$ Hamiltonians whose eigenvalues have constant multiplicities and do not cross on $[0,1]$, we construct a unitary stochastic adiabatic intertwiner driven by the same Wiener process as the exact propagator. A commutator argument, Itô calculus, and Doob's maximal inequality yield an explicit bound for the expected supremum of their squared operator-norm difference on a selected initial eigenspace. Under uniform coefficient bounds, the physical noise scaling $\alpha =\sigma T^{-\gamma }$, where $T$ is the total evolution time and $\sigma \ge 0$ and $\gamma \ge 0$ are fixed, gives maximal mean-square error $O(T^{-2}+{\sigma }^4{T}^{-4\gamma }+{\sigma }^2{T}^{-(2\gamma +1)})$. Consequently, adiabatic convergence holds for every $\gamma >0$ as $T\to \infty$, while $\gamma \ge 1/2$ recovers the deterministic $O(T^{-1})$ bound for the root-mean-square maximal error. For fixed nonzero physical noise amplitude, the estimate leaves convergence unresolved.