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, $C^2$ 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 $\theta$, whereas prime-dimensional stabilizer Rényi entropies scale as ${\theta }^{2\alpha }$, ${\theta }^2\ln (1/\theta )$, and ${\theta }^2$ for fixed $0<\alpha <1$, $\alpha =1$, and $1<\alpha <\infty$, 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.

arXiv comment72 pages, 1 figure. Substantially revised and expanded, with a new title. Develops a general framework for quantum-trajectory ergodicity and singular stationary response, including affine tangent limits and rare-excursion occupation measures