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

TL;DR

This work analyzes how the quantum Fisher information matrix (QFIM) signals criticality and non-Markovianity in the spin-boson model by computing static and dynamical QFIM elements with numerically exact tensor-network methods. It reveals a non-perturbative divergence of $F_{\alpha\alpha}$ as $\alpha\to0$, and a pronounced peak near the Berezinskii-Kosterlitz-Thouless transition at $\alpha_c\approx1.03$, indicating enhanced sensing at criticality. The dynamical QFIM exhibits oscillations with a frequency twice that of the spin observable in the coherent regime, and positive QFI flow signals non-Markovian information backflow that vanishes in the incoherent regime, with these features persisting under pure dephasing. Collectively, the results establish QFIM as a sensitive probe of criticality and non-Markovian dynamics in open quantum systems, offering insights for metrology and quantum sensing in noisy environments.

Abstract

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.

Figures (10)

• Figure 1: QFIM element $F_{\alpha\alpha}$ (a), $\langle\sigma_z\rangle$ (b), and von Neumann entropy (c) of the Hamiltonian GS as functions of spin-bath coupling $\alpha$ for several values of the magnetic field $h$ (in units of $\Delta$). The dashed line in panel c indicates the maximum possible value $S_{max}=\ln2\approx0.69$ for a TLS. In all the plots, we set the spin-bath dephasing coupling $\kappa=0$.
• Figure 2: GS QFIM element $F_{\alpha\alpha}$ (in logarithmic scale) of the generalized SBM as a function of the spin-bath coupling $\alpha$, for $h=10^{-6}$ (in units of $\Delta$) and different values of the dephasing coupling $\kappa$. The $\kappa=0$ curve corresponds to the $h=10^{-6}$ case in Fig. \ref{['Static_QFI_new']} (a): its minimum becomes visible only in logarithmic scale.
• Figure 3: QFIM element $F_{\alpha\alpha}$, calculated with methods based on MPS and Lindblad master equation, as a function of time for spin-bath coupling $\alpha=0.1$ (a), $\alpha=0.25$ (b) and $\alpha=0.5$ (c), at the spin-bath coupling $\kappa=0$. The dashed horizontal line is the ground state QFI value calculated with the DMRG method: $F_{\alpha\alpha}^{\alpha=0.1}=7.7$, $F_{\alpha\alpha}^{\alpha=0.25}=3.15$, and $F_{\alpha\alpha}^{\alpha=0.5}=1.21$.
• Figure 4: QFIM elements $F_{\alpha\Delta}$ (in units of $1/\Delta$) (a), and $F_{\Delta\Delta}$ (in units of $1/\Delta^2$) (b) as functions of spin-bath coupling $\alpha$ for various values of $h$ (in units of $\Delta$) at $\kappa=0$.
• Figure 5: Extrapolation of the critical coupling $\alpha_c$ from the behavior of the QFI $F_{\alpha\alpha}$ (a) and $F_{\Delta\Delta}$ (b) as a function of the magnetic field $h$ (in units of $\Delta$). We also show the resulting power-law fit in Eq. \ref{['Fit']}.