Quantum Fisher information as a witness of non-Markovianity and criticality in the spin-boson model

The quantum Fisher information, the quantum analogue of the classical Fisher information, is a central quantity in quantum metrology and quantum sensing due to its connection to parameter estimation and fidelity susceptibility. Using numerically exact methods applied to a paradigmatic open quantum system, the spin-boson model, we calculate both static and dynamical quantum Fisher information matrix elements with respect to spin-bath couplings and magnetic field strengths. As the spin-bath interaction increases, we first show that the coupling-coupling matrix elements relative to the ground state of the Hamiltonian are linked to the entanglement growth and signal the Berezinskii-Kosterlitz-Thouless quantum phase transition through their non-monotonic behavior. We also point out that the static quantum Fisher information exhibits a non-perturbative behavior in the zero-coupling limit, which we justify with an analytic argument. Furthermore, we demonstrate that the time-dependent matrix elements can reveal non-Markovian effects as well as the transition from the coherent to incoherent regime at the Toulouse point, remaining robust under pure dephasing noise. Non-monotonic signatures of the quantum Fisher information matrix reflect changes in quantum resources such as entanglement and coherence, quantify non-Markovian behavior, and enable criticality-enhanced quantum sensing, thereby shedding light on key features of open quantum systems.

💡 Research Summary

This paper investigates how the quantum Fisher information (QFI) and its matrix form, the quantum Fisher information matrix (QFIM), can serve as a comprehensive probe of both static and dynamical properties of an open quantum system, focusing on the paradigmatic spin‑boson model (SBM). The authors employ numerically exact tensor‑network techniques—density‑matrix renormalization group (DMRG) for ground‑state properties and matrix‑product‑state (MPS) time‑evolution (WI2 algorithm) for dynamics—supplemented by analytical Lindblad master‑equation calculations for comparison. The SBM consists of a two‑level system (spin‑½) coupled to an Ohmic bosonic bath with spectral density J(ω)=α ω θ(ω_c−ω) and includes both amplitude‑damping (σ_z) and pure‑dephasing (σ_x) channels, the latter controlled by a dimensionless strength κ.

In the static sector, the QFIM elements with respect to the spin‑bath coupling α and the transverse field Δ (F_{αα}, F_{αΔ}, F_{ΔΔ}) are computed as functions of α for several small longitudinal fields h. All three elements display a pronounced peak near the Berezinskii‑Kosterlitz‑Thouless (BKT) quantum phase transition at a critical coupling α_c≈1.03 (for the chosen cutoff ω_c=10Δ). As h→0⁺ the peak sharpens and its position converges to α_c, indicating that the ground‑state wavefunctions on opposite sides of the transition become nearly orthogonal, which mathematically drives the QFIM to diverge. The authors extract α_c directly from the peak positions, obtaining α_c=1.03±0.03, in excellent agreement with previous literature.

A second, less expected feature emerges in the ultra‑weak coupling limit α→0. Here the coupling‑coupling element F_{αα} diverges as 1/α, a non‑perturbative behavior that the authors confirm analytically by expanding the ground‑state expectation values ⟨σ_x⟩≈1+αc (with c≈−1.48) and using the general QFIM formula. This divergence signals that even infinitesimal system‑bath interactions can render the state extremely sensitive to variations of α, a property that could be exploited for ultra‑weak‑coupling metrology.

The static analysis also reveals a monotonic decrease of F_{αα} as α grows from the weak to intermediate regime, which correlates with an increase of the spin’s von Neumann entropy S(ρ_spin). As the spin becomes more entangled with the bath, its ability to encode information about α diminishes, illustrating a trade‑off between entanglement and parameter sensitivity. Introducing a finite dephasing strength κ reduces the height of the critical peak (since κ acts like an additional longitudinal field) but does not eliminate the 1/α divergence, demonstrating robustness of the QFIM signatures against pure‑dephasing noise.

Turning to dynamics, the authors study the time‑dependent QFIM element F_{αα}(t) for three representative couplings: α=0.1, 0.25, and 0.5 (the Toulouse point). They compare exact MPS results with those obtained from a Lindblad master equation, which is only valid in the weak‑coupling, Markovian regime. In the coherent regime (α<0.5) the exact QFIM exhibits sustained oscillations, whereas the Lindblad prediction quickly damps to a stationary value. The oscillation frequencies differ from those of the spin observable ⟨σ_z⟩, indicating that the QFIM captures memory effects (non‑Markovianity) that are invisible in simple population dynamics. For α>0.5 the system enters an incoherent regime; the QFIM then decays monotonically without revivals, reflecting the loss of information backflow. The authors quantify non‑Markovianity via the QFI flow (time derivative of the QFIM) and show that positive flow intervals correspond precisely to the oscillatory revivals in the coherent regime.

Additional analyses in the appendices explore (i) the extraction of the critical coupling from various QFIM components, (ii) the analytic weak‑coupling expansion confirming the 1/α divergence, (iii) the behavior of the Bloch‑vector z‑component under dephasing, (iv) the detailed relationship between QFI flow and established non‑Markovianity measures, and (v) the limitations of the Lindblad approach, which fails to reproduce the stationary QFIM in the strong‑coupling limit.

Overall, the paper demonstrates that the QFIM provides a unified metric for (a) detecting quantum criticality (via divergent static QFI at the BKT transition), (b) diagnosing non‑Markovian dynamics (through oscillatory QFI flow), and (c) assessing the impact of environmental noise (robustness against pure dephasing). The non‑monotonic behavior of the QFIM links directly to underlying quantum resources such as entanglement and coherence, suggesting that criticality‑enhanced QFI could be harnessed for ultra‑sensitive magnetic‑field sensing. The work thus positions the quantum Fisher information matrix as a powerful, experimentally relevant tool for probing and exploiting the rich physics of open quantum systems.