Stochastic Quantum Information Geometry and Speed Limits at the Trajectory Level

TL;DR

This work introduces the Conditional Quantum Fisher Information (CQFI) to extend quantum metrology from ensemble-averaged quantities to single quantum trajectories, unifying quantum information geometry with stochastic thermodynamics. CQFI is defined via the Symmetric Logarithmic Derivative and conditioned on measurement outcomes, and it decomposes into incoherent, coherent, and a trajectory-specific cross-term that can be negative, vanishing only after ensemble averaging. By applying a trajectory-level framework, the authors define stochastic thermodynamic length and action and derive quantum speed limits valid for individual realizations, validated through quantum-jump simulations of driven thermal qubits and Gaussian-state force sensing. The results reveal that trajectory-level bounds can be tighter than ensemble counterparts and identify destructive interference as a purely quantum, trajectory-local witness, with potential applications in adaptive metrology, real-time feedback, and efficient sampling of rare high-information trajectories.

Abstract

Standard quantum metrology relies on ensemble-averaged quantities, such as the Quantum Fisher Information (QFI), which often mask the fluctuations inherent to single-shot realizations. In this work, we bridge the gap between quantum information geometry and stochastic thermodynamics by introducing the Conditional Quantum Fisher Information (CQFI). Defined via the Symmetric Logarithmic Derivative, the CQFI generalizes the classical stochastic Fisher information to the quantum domain. We demonstrate that the CQFI admits a decomposition into incoherent (population) and coherent (basis rotation) contributions, augmented by a transient interference cross-term absent at the ensemble level. Crucially, we show that this cross-term can be negative, signaling destructive interference between classical and quantum information channels along individual trajectories. Leveraging this framework, we construct a stochastic information geometry that defines thermodynamic length and action for single quantum trajectories. Finally, we derive fundamental quantum speed limits valid at the single-trajectory level and validate our results using the quantum jump unraveling of a driven thermal qubit.

Figures (5)

• Figure 1: Sketch of the conditional quantum Fisher information for the adopted example in this work. Panel (a) shows a sketch of a qubit in contact with a thermal bath, submitted to a driving potential. When the system is monitored, it produces stochastic outcomes for each realization. Panel (b) depicts the results for trajectory level quantum Fisher information decomposed into the contributions from population changes in blue, the contributions from unitary rotations of the eigenbasis in red, and the cross contributions from the interference between both channels in green. Panel (c) shows the results from the averaged QFI on ensemble level, where the cross contributions vanish.
• Figure 2: Decomposition and validation of the CQFI. (a) The agreement of the average CQFI $\langle f_Q\rangle$ (in blue) with the QFI $\mathcal{F}_Q$ (in orange dashed), with relative error $\le 1\%$. (b) Decomposition of $\langle f_Q\rangle$ into populations $\langle f_Q^{{\rm IC}}\rangle$, coherent $\langle f_Q^{{\rm C}}\rangle$, and cross $f_Q^{{\rm X}}$ contributions. The coherence term dominates at longer times, highlighting the quantum nature of the dynamics. (c) Decomposition of $\mathcal{F}_Q$ (black dashed) into populations $\mathcal{F}_Q^{{\rm IC}}$ (in red) and coherences $\mathcal{F}_Q^{{\rm C}}$ (in blue) contributions, where the cross term is strictly zero. (d) Trajectory level fluctuations of the CQFI (thin lines) compared against the ensemble averages $\langle f_Q\rangle$ (in black) and $\mathcal{F}_Q$ (in orange, dashed).
• Figure 3: Verification of information geometry and speed limits. (a) Convergence of the average stochastic action $\langle j\rangle(t)$ to the ensemble-level statistical action $\mathcal{J}(t)$. (b) Comparison of speed limits at ensemble and averaged-trajectory level. The bound derived for ensemble QFI is consistently higher than or equal to average of single-trajectories bounds, respecting the expected hierarchy. (c) Convergence of $\langle f_Q^{{\rm X}}\rangle$ toward zero as the number of trajectories increases, for different time steps. (d) Verification of the trajectory-level speed limit inequality for a subset of the $N_{{\rm trajs}} = 5\times10^4$ simulated trajectories.
• Figure 4: Speed limits for the rate of change of the Hamiltonian expectation value. (a) Ensemble-level speed limit [cf. Eq.\ref{['eq:Avg_QSLs_Obs_QFI']}]. (b) Trajectory-level speed limit [cf. Eq.\ref{['eq:Trajs_QSLs_Obs_QFI']}]. Since the Hamiltonian is time-independent in the rotating frame, the rate of change is zero for all times. Consequently, the speed limit inequalities are trivially satisfied, with the geometric bounds (RHS) remaining strictly positive and bounding the zero velocity (LHS) throughout the evolution.
• Figure 5: Time evolution of the CQFI and QFI for force sensing in dislocated Gaussian states. Due to the Gaussian nature of the state and measurement, the CQFI coincides with the QFI. Both quantities scale quadratically with time, $\mathcal{F}_{Q} \sim t^2$.