Probability bounds on trajectory-level exceedances of quantum speed limits under continuous-measurement feedback

Abstract

We investigate the probability that a conditioned quantum trajectory exceeds an ensemble-derived quantum speed limit (QSL) velocity under continuous measurement-based feedback. Using the finite-time average fidelity velocity, we derive an upper bound on the exceedance probability in terms of its mean and variance. We further obtain state-independent bounds from uniform constraints on the fidelity drift and diffusion, including an exponential concentration bound. For a feedback-controlled qubit, we derive explicit model-specific bounds and show that they are nontrivial in a short-time regime. We also identify a finite-time regime in which the exceedance probability vanishes exactly due to the restricted support of the conditioned dynamics. Monte Carlo simulations confirm the analytical bounds and illustrate their tightness.

The paper asks how likely a continuously monitored quantum trajectory with measurement-based feedback is to exceed a quantum-speed-limit reference derived from ensemble evolution. It studies the finite-time initial-state fidelity velocity $v_c=(1-F_\tau^c)/\tau$ and the reference $v_{\rm QSL}=\tau^{-1}\int_0^\tau\|\dot\rho_t\|_\infty\,dt$. Such exceedances compare conditioned and ensemble dynamics; they are not violations of a trajectory-level QSL. Applying Cantelli’s inequality gives $\Pr(v_c\geq v_{\rm QSL})\leq\sigma_v^2/[\sigma_v^2+(v_{\rm QSL}-\mathbb E v_c)^2]$, provided the threshold strictly exceeds the mean. Uniform fidelity-drift and diffusion bounds, $-A\leq A_{\max}$ and $B^2\leq B_{\max}^2$, yield the distribution-independent estimate $\min\{B_{\max}^2/(B_{\max}^2+\tau\delta^2),\exp[-\tau\delta^2/(2B_{\max}^2)]\}$ for $\delta=v_{\rm QSL}-A_{\max}>0$, using Itô isometry and an exponential martingale. Spectral-width estimates provide explicit constants for bounded feedback. The example is an ideally monitored qubit with $L=\sqrt\kappa\sigma_z$ and feedback Hamiltonian $-\alpha\kappa x_c\sigma_y$. Pure-state Bloch geometry gives $B_{\max}^2=\kappa/4$ and an explicit optimized drift bound. The concentration condition holds at sufficiently short times. Moreover, $x_c>0$ at every finite time implies $F_\tau^c>1/2$, so the exceedance probability is exactly zero whenever $v_{\rm QSL}\geq1/(2\tau)$. Monte Carlo comparisons use $5\times10^4$ trajectories and dimensionless timestep $10^{-3}$. The moment bound requires trajectory statistics and becomes trivial when the ensemble reference is saturated. The state-independent concentration bounds require a positive drift–threshold gap. Feedback generally prevents closed ensemble dynamics, so evaluating the reference can still require simulation.