Quantum Adiabatic Theorem under Wiener Noise:Maximal Mean-SquareEstimates

Abstract

We derive maximal mean-square adiabatic estimates for finite-dimensional stochastic Schrödinger equations with Hamiltonian-proportional Wiener noise. For deterministic C² 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 α=σT^−γ, where T is the total evolution time and σ≥0 and γ≥0 are fixed, gives maximal mean-square error O(T⁻²+σ⁴T^−4γ+σ²T^−(2γ+1)). Consequently, adiabatic convergence holds for every γ>0 as T→∞, while γ≥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.

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BibTeX RIS

BibTeX

@misc{caldera2026quantum,
  title = {{Quantum Adiabatic Theorem under Wiener Noise:Maximal Mean-SquareEstimates}},
  author = {A. M. H. H. Caldera and R. P. K. DeSilva and K. K. W. A. S. Kumara and B. P. W. Fernando},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06029},
  eprint = {2611.06029},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06029},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Quantum Adiabatic Theorem under Wiener Noise:Maximal Mean-SquareEstimates
AU  - A. M. H. H. Caldera
AU  - R. P. K. DeSilva
AU  - K. K. W. A. S. Kumara
AU  - B. P. W. Fernando
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06029
UR  - https://arxiv.org/abs/2611.06029
AB  - 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 α=σT^−γ, where T is the total evolution time and σ≥0 and γ≥0 are fixed, gives maximal mean-square error O(T^−2+σ^4T^−4γ+σ^2T^−(2γ+1)). Consequently, adiabatic convergence holds for every γ>0 as T→∞, while γ≥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.
ER  -

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