Quantum Trajectories: Ergodicity, Perturbative Dynamics, and Stationary Response
Abstract
Quantum trajectories capture measurement-conditioned statistics beyond ensemble averages. For finite-dimensional Kraus dynamics, complete-reset and overlap conditions yield a unique invariant law with explicit geometric total-variation contraction. Under finite-reference mixing and quantitative transverse contraction, C2 Kraus perturbations yield two complementary stationary limits: an affine tangent law for local fluctuations and an occupation measure for rare excursions. These limits give leading observable-response formulas, including an excursion contribution at quadratic order; both limits and coefficients depend only on reference Kraus operators and their first derivatives. For Clifford-driven monitored trajectories, group structure verifies the general hypotheses and yields geometric Wasserstein mixing uniformly over intermediate unitaries at fixed dimensions, an equivalence between uniform total-variation ergodicity and a finite nonzero rank-one Kraus product for each fixed unitary, and exact stationary tangent energies. Under partial measurement, tangent dynamics have almost surely finite memory with a sharp exponential tail; every C2 state observable has vanishing first-order stationary response. For weak Hamiltonian perturbations, odd-prime Gross--Wigner mana responds linearly in the perturbation strength θ, whereas prime-dimensional stabilizer Rényi entropies scale as θ2α, θ2ln(1/θ), and θ2 for fixed 0<α<1, α=1, and 1<α<∞, respectively. Under partial measurement, the leading coefficients are strictly positive precisely for Hamiltonians outside the real span of the measurement projectors; on this quotient, the mana coefficient defines a norm and the order-one logarithmic coefficient an explicit positive-definite quadratic form.
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BibTeX
@misc{zhen2026quantum,
title = {{Quantum Trajectories: Ergodicity, Perturbative Dynamics, and Stationary Response}},
author = {Guocheng Zhen and Xuanrong Yang and Chengkai Zhu and Ranyiliu Chen and Xin Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.13470},
eprint = {2606.13470},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.13470},
}
RIS
TY - UNPB
TI - Quantum Trajectories: Ergodicity, Perturbative Dynamics, and Stationary Response
AU - Guocheng Zhen
AU - Xuanrong Yang
AU - Chengkai Zhu
AU - Ranyiliu Chen
AU - Xin Wang
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2606.13470
UR - https://arxiv.org/abs/2606.13470
AB - Quantum trajectories capture measurement-conditioned statistics beyond ensemble averages. For finite-dimensional Kraus dynamics, complete-reset and overlap conditions yield a unique invariant law with explicit geometric total-variation contraction. Under finite-reference mixing and quantitative transverse contraction, C^2 Kraus perturbations yield two complementary stationary limits: an affine tangent law for local fluctuations and an occupation measure for rare excursions. These limits give leading observable-response formulas, including an excursion contribution at quadratic order; both limits and coefficients depend only on reference Kraus operators and their first derivatives. For Clifford-driven monitored trajectories, group structure verifies the general hypotheses and yields geometric Wasserstein mixing uniformly over intermediate unitaries at fixed dimensions, an equivalence between uniform total-variation ergodicity and a finite nonzero rank-one Kraus product for each fixed unitary, and exact stationary tangent energies. Under partial measurement, tangent dynamics have almost surely finite memory with a sharp exponential tail; every C^2 state observable has vanishing first-order stationary response. For weak Hamiltonian perturbations, odd-prime Gross--Wigner mana responds linearly in the perturbation strength θ, whereas prime-dimensional stabilizer Rényi entropies scale as θ^2α, θ^2ln(1/θ), and θ^2 for fixed 0<α<1, α=1, and 1<α<∞, respectively. Under partial measurement, the leading coefficients are strictly positive precisely for Hamiltonians outside the real span of the measurement projectors; on this quotient, the mana coefficient defines a norm and the order-one logarithmic coefficient an explicit positive-definite quadratic form.
ER -