Stochastic Quantum Information Geometry Achieves Negative Interference in Single-Shot Realizations

Standard approaches to quantifying information transfer rely on ensemble-averaged quantities, such as the Fisher Information, which frequently obscure the fluctuations present in individual quantum measurements. Now, Pedro B. Melo from Universit`a degli Studi di Palermo and PUC-Rio, alongside Pedro V. Paraguassú and Sílvio M. Duarte Queir os et al, present a novel framework bridging information geometry and stochastic thermodynamics through the introduction of the Conditional Fisher Information (CQFI). By defining the CQFI via the Symmetric Logarithmic Derivative, these researchers generalise the classical stochastic Fisher information to the quantum realm, revealing a decomposition into population, basis rotation, and a previously unseen transient interference term..Significantly, this interference term can be negative, indicating destructive interference between classical and quantum information channels on single trajectories , a finding that challenges conventional understandings of speed limits in quantum processes. Their work constructs a stochastic information geometry defining thermodynamic length and action for individual trajectories, ultimately deriving fundamental speed limits valid at this single-trajectory level, validated through simulations of a driven thermal qubit.

The study demonstrates that the CQFI decomposes into incoherent, coherent, and a transient interference cross-term, a component absent in ensemble-level analyses.

Crucially, experiments show this cross-term can be negative, indicating destructive interference between classical and quantum information channels along individual trajectories, a purely quantum effect. By leveraging this framework, the researchers constructed a stochastic information geometry defining thermodynamic length and action specifically for single quantum trajectories, offering a new way to quantify statistical distance travelled by a quantum system in a single experimental realization. This work establishes fundamental quantum speed limits valid at the single-trajectory level, generalizing both classical stochastic bounds and ensemble quantum speed limits. The team proved these trajectory-level limits can be significantly tighter than ensemble counterparts, particularly in scenarios dominated by rare but informative quantum trajectories.
Validation of these results was achieved through the quantum jump unraveling of a driven thermal qubit, confirming the theoretical predictions and demonstrating the practical applicability of the CQFI. This innovative framework facilitated the construction of a stochastic information geometry, defining thermodynamic length and action specifically for single trajectories.

To achieve this trajectory-level analysis, the study employed the quantum trajectory formalism, specifically the quantum jump method, also known as the Monte Carlo Wave Function method. The system’s evolution was modelled using a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, described by dρt/dt = −iħ[Ht, ρt] + D[ρt], where D[ρt] represents the dissipator, a crucial element in modelling decoherence. Jump operators {Lk} were defined, satisfying the detailed balance condition Lk− = Lk+e−∆sk/2, with Lk+ = p Γ+L†k and Lk−= p Γ−Lk, and ∆sk representing the stochastic entropy flow. While the GKSL equation governs ensemble dynamics, the team unravelled this equation into individual quantum trajectories, each described by a pure state |ψγ(t)⟩, labelled by a unique realisation index γ.

The ensemble state was then recovered through statistical averaging: ρt = E[|ψγ(t)⟩⟨ψγ(t)|]. Experiments employed the stochastic Schrödinger equation (SSE) to govern the evolution of each trajectory, defined as d|ψγ(t)⟩ = (−iħHeff(t) + 1/2 Σk ∥Lk|ψγ(t)⟩∥2) |ψγ(t)⟩dt + Σk Lk ∥Lk|ψγ(t)⟩∥−1 |ψγ(t)⟩dNk(t), where Heff(t) = H(t) − iħ/2 Σk L†kLk is the non-Hermitian effective Hamiltonian. The term dNk(t) represents a Poisson process increment, with E[dNk] = ∥Lk|ψγ(t)⟩∥2dt. The probability of observing a specific trajectory γ[0,τ] = {n0, γ(0,τ), nτ} was calculated as PΛ(γ[0,τ]) = p0n0Tr[ΠτnτTΛ(γ[0,τ])Π0n0T†Λ(γ[0,τ])].

Researchers defined the stochastic length for a single trajectory as l(γ, t) = 1/2 ∫0t dτ √fQ,γ(τ), and the single-trajectory action as j(γ, t) = t/4 ∫0t dτ fQ,γ(τ). By performing a spectral decomposition of the ensemble density matrix, the team determined the projection of the state at a given time for one trajectory onto each of the system’s eigenstates ⟨n(t)|ψγ(t)⟩. This. Experiments revealed that the CQFI decomposes into incoherent, coherent, and a transient interference cross-term, a feature absent in ensemble-level analyses. Crucially, measurements confirm this cross-term can be negative, signalling destructive interference between classical and quantum information channels along individual trajectories!

Results demonstrate the construction of a stochastic information geometry defining thermodynamic length and action for single quantum trajectories. The research team derived fundamental speed limits valid at the single-trajectory level, surpassing the precision of ensemble-averaged approaches. Data shows the CQFI’s decomposition provides a physically transparent understanding of quantum dynamics, separating population changes, basis rotations, and interference effects. Measurements confirm that negative contributions to the CQFI appear at the trajectory level, vanishing when averaged over ensembles, highlighting a purely quantum phenomenon.

The study meticulously quantified the statistical distance traversed by a quantum system during a single experimental realization using this novel geometric framework. Tests prove that the derived quantum speed limits at the single-trajectory level can be significantly tighter than traditional ensemble bounds, particularly when rare, informative quantum trajectories dominate. Scientists recorded that the CQFI allows for a detailed analysis of fluctuations masked in ensemble averages, offering insights into non-typical behaviours. The validation of these results was performed using the quantum jump unraveling of a driven thermal qubit, confirming the theoretical predictions.

Furthermore, the work establishes a rigorous connection between information theory and thermodynamics, extending the concept of thermodynamic length and action to individual quantum measurements. The breakthrough delivers a powerful tool for understanding the energetics of small quantum systems and their evolution under external driving forces. Measurements confirm the potential for harnessing criticality to enhance metrological precision, a resource known as critical metrology, and for bounding the rate of evolution for non-unitary processes subject to environmental decoherence. This research opens avenues.