Dynamic Quantum Fisher Information

• Dynamic quantum Fisher information is a measure that quantifies the instantaneous sensitivity of quantum systems to parameter changes, capturing key features like entanglement and distinguishability.
• It provides rigorous bounds for quantum metrology by linking system evolution, non-Markovian behavior, and phase transitions to precision limits.
• The framework extends to multi-parameter and open system scenarios, enabling detailed analysis of decoherence, resource quantification, and critical dynamics.

Dynamic quantum Fisher information (QFI) quantifies the instantaneous sensitivity of a quantum system—pure or mixed, closed or open—to changes in physically relevant parameters as the system evolves in time. It provides a rigorous operational measure of distinguishability in parameter space, bounds quantum-enhanced estimation error, and directly diagnoses dynamical entanglement, coherence, non-Markovianity, and phase transitions in unitary and dissipative many-body evolution. The full dynamical QFI framework encompasses time-dependent state QFI, channel QFI, their generalization to multi-parameter and open system settings, and their functional role in quantum metrology bounds, resource theory, quantum thermodynamics, and critical phenomena.

1. Fundamental Definitions and Mathematical Structure

Dynamic quantum Fisher information is constructed from a family of parameter-dependent quantum states $\rho(\theta, t)$ or channels $\mathcal{E}_{\theta, t}$. For a mixed state evolving under any protocol (unitary, CPTP, Lindbladian), the symmetric logarithmic derivative (SLD) $L_\theta(t)$ is defined as the Hermitian solution to: $$\partial_\theta \rho(\theta, t) = \frac{1}{2}(L_\theta(t) \rho(\theta, t) + \rho(\theta, t) L_\theta(t))$$ The (single-parameter) dynamic QFI is given by: $$F_Q[\rho(\theta, t)] = \mathrm{Tr}[\rho(\theta, t)\,L_\theta^2(t)]$$ For pure states, it reduces to $F_Q = 4\,\mathrm{Var}_{\Psi(t)}[O]$ for a generator $O$. For a vector of parameters $\vec{x}$, the quantum Fisher information matrix (QFIM) generalizes as: $$F_{\mu\nu}(t) = \frac{1}{2}\,\mathrm{Tr}\left[\rho(t)\{L_\mu(t), L_\nu(t)\}\right]$$ where $L_\mu(t)$ solves $$\partial_{x_\mu}\rho(t) = \frac{1}{2}(L_\mu(t)\rho(t) + \rho(t)L_\mu(t))$$ (Parlato et al., 22 Aug 2025).

For channels parameterized by $\theta$, dynamic QFI is: $$F_Q(\mathcal{E}_\theta) = \sup_{\rho}\,F_Q[\mathcal{E}_\theta(\rho)]$$ i.e., the maximal achievable QFI over all probe states (Tan et al., 2021).

2. Dynamical Evolution: Closed and Open Systems

In closed systems, the evolution is unitary: $\rho(t) = U(t)\rho_0 U^\dagger(t)$; QFI dynamics track spread in the generator basis. For a time-dependent unitary family $U(\theta, t)=e^{-i t H(\theta)}$ (e.g. SU(2) processes), the optimal QFI splits into quadratic-in-time and oscillatory parts: $$F_{\mathrm{max}}(t) = 4\left\| (d r/d\theta)\,\mathbf{e}_r \right\|^2 t^2 + \frac{16}{r^2}\left\| r\,d\mathbf{e}_r/d\theta \right\|^2 \sin^2\left(\frac{r t}{2}\right)$$ This decomposition has clear geometric meaning: the time-square term measures generator norm variation; the oscillatory term, generator axis rotation (Jing et al., 2015).

For open systems governed by a Lindblad or Liouvillian master equation: $$\frac{d}{dt}\rho(\theta, t) = \mathcal L_\theta[\rho(\theta, t)], \qquad \rho(\theta, t) = e^{t\,\mathcal L_\theta}[\rho_0]$$ the dynamic QFI is upper-bounded by closed-form expressions involving the non-Hermitian SLD, e.g. for Hamiltonian estimation,

$$F_{\mathrm{ext}}[\rho(\theta_0, t)] = 4 t^2 \,\mathrm{Var}_{\rho(t)}(H)$$

For dissipative parameters, it involves traces over Lindblad operators and their action on $\rho(t)$, yielding rich time dependence (quadratic, exponential, oscillatory) set by the Liouvillian spectrum (Alipour et al., 2014, Peng et al., 2023).

3. Scaling, System Size, and Critical Behavior

Dynamic QFI provides critical scaling information:

• Heisenberg limit: For multipartite systems, $F_Q\sim N^2$ witnesses genuine multipartite entanglement and sets the ultimate quantum metrological bound (Gietka et al., 2018, Mumford et al., 8 Jul 2025).
• Standard quantum limit/shot noise: Linear scaling $F_Q\sim N$ arises in separable or decohered regimes.
• Crossover: In open systems, dynamic QFI interpolates between quantum ($N^2$) and classical ($N$) regimes, with the threshold set by decoherence strength and interrogation time—see $N^*=c_1/c_2$ in (Alipour et al., 2014).

At phase transitions, time-averaged QFI displays universal nonanalytic behavior, serving as an order parameter for dynamical quantum phase transitions (DQPTs) (Mumford et al., 8 Jul 2025). In dissipative many-body systems, dynamical QFI also reveals transitions from Markovian (monotonic loss) to non-Markovian (information backflow, oscillatory or even revival) behavior (Parlato et al., 22 Aug 2025, Xing et al., 2021, Scandi et al., 2023).

4. Entanglement, Metrology, and Resource Quantification

Dynamic QFI detects and quantifies multipartite entanglement—$F_Q > N$ (for N qubits) implies at least $k$-partite entanglement, with $k \sim F_Q/N$ (Gietka et al., 2018). QFI underlies the quantum Cramér–Rao bound for parameter estimation: $\delta\theta \geq 1/\sqrt{M F_Q}$ with $M$ repetitions. In dynamic protocols (e.g., Dicke, Lipkin–Meshkov–Glick, spin-boson models), ergodic or chaotic evolution can exponentially accelerate the growth of useful entanglement, lowering the time scale for reaching the Heisenberg limit from $\sim\sqrt{N}$ to $\sim\log N$ (Gietka et al., 2018).

In resource theories, dynamic QFI (channel or state) quantitatively distinguishes resourceful channels and states from free ones, establishing Fisher information as a universal resource witness (Tan et al., 2021).

5. Markovianity, Non-Markovianity, and Information Flow

The instantaneous time derivative of dynamic QFI, or its geometric generalization (intrinsic density flow, IDF), is a sharp witness of non-Markovianity:

• For CP-divisible (Markovian) evolution, $\partial_t F_Q(t) \leq 0$.
• Temporary $\partial_t F_Q(t) > 0$ signals non-Markovian information backflow from environment to system.

IDF generalizes to multi-parameter manifolds; its sign and magnitude express quantifiable loss or regaining of local distinguishability in time (Xing et al., 2021, Scandi et al., 2023). These concepts are analytically and numerically verified in models such as spin-boson, Yang-Baxterized spin chains, and qubit dephasing (Parlato et al., 22 Aug 2025, Duran, 2020).

6. Decoherence Channels and Analytic QFI Evolution

Explicit dynamic QFI formulas have been derived for memoryless decoherence channels (phase-damping, depolarizing, amplitude-damping) for both single-qubit and multi-qubit (X-state, GHZ, Werner) inputs (Zhong et al., 2012, Naimy et al., 2 Dec 2024, Abouelkhir et al., 2022). Key findings:

• Phase-damping and phase-flip noise preserve residual QFI even after entanglement vanishes ("entanglement sudden death"), maintaining metrological advantage.
• Depolarizing channels eradicate both entanglement and QFI, restoring classical scaling.
• Time-dependent block-decomposition and affine Bloch representations facilitate analytic expressions for general qudit systems (Zhong et al., 2012, Naimy et al., 2 Dec 2024).

7. Dynamical QFI in Many-Body and Nonequilibrium Scenarios

Recent work demonstrates that local nonequilibrium "kicking" (transient local perturbations) amplifies subsystem QFI from $O(L)$ (linear in block length $L$) to $O(L^2)$ (quadratic scaling), transiently producing macroscopic entangled states within compact blocks. These enhancements are tunable via protocol structure (single/local/global kicks, periodic driving), subject to a characteristic timescale for entanglement generation and subsequent decay (Ferro et al., 27 Mar 2025). The connection between QFI and Wigner–Yanase–Dyson skew information gives computational access to dynamical multipartite entanglement in large subsystems.

Table: Dynamic QFI Scaling and Metrological Bounds

Regime	QFI Scaling	Parameter Estimation Bound
Separable/shot noise	$F_Q \lesssim N$	$\delta\theta \sim 1/\sqrt{N}$
Maximally entangled	$F_Q \sim N^2$	$\delta\theta \sim 1/N$
Decohered/open (large $t$)	$F_Q \sim N$ (eventually)	Standard classical limit
Dynamic enhancement (local kicks, chaotic)	$F_Q \sim L^2$ (for block size $L$ in optimal window)	Macroscopic multipartite entanglement for transient time $t^* \sim L/v_M$

Critical points, ergodicity, and non-Markovianity yield nontrivial time and system-size scaling for QFI in many-body protocols.

Applications and Experimental Realizations

Dynamic QFI protocols inform quantum metrology, sensing, thermodynamics, and quantum control. Criticality (e.g., near quantum phase transitions) enhances sensitivity (divergent QFI) (Parlato et al., 22 Aug 2025, Mumford et al., 8 Jul 2025). Decoherence-resilient metrological strategies leverage channels and time windows where QFI remains robust. Adaptive measurement routines can exploit analytic expressions for QFI under noise to maximize achievable precision and optimize timing (Alipour et al., 2014, Naimy et al., 2 Dec 2024). Non-Markovianity witnesses facilitate experimental discrimination of environmental memory and retrodiction (Scandi et al., 2023, Xing et al., 2021).

Open Problems and Future Directions

Challenges include exact computation of maximal multi-parameter QFIMs in noisy, strongly interacting systems; extension of dynamic QFI beyond quantum mechanics to generalized probabilistic theories; and experimental implementation of protocols that exploit dynamical enhancement, criticality, and non-Markovianity for robust quantum sensing (Chen et al., 2017, Scandi et al., 2023). The relation of QFI to generalized skew information and recovery maps continues to unify operational and geometric perspectives.

Dynamic quantum Fisher information serves as a unifying metric for real-time quantum estimation, entanglement dynamics, resource identification, non-Markovianity detection, and criticality witnessing, with rigorous analytic and numerical foundations in diverse quantum many-body contexts.