Finite-Size Effects in Quantum Metrology at Strong Coupling: Microscopic vs Phenomenological Approaches

The gist The study investigates the ultimate precision limits of a spin chain, strongly coupled to a heat bath, for measuring general parameters and reports results for magnetometry and thermometry <ref:2507.19994#pg2>.

Model and Effective Hamiltonian

The research employs the one-dimensional spin-1/2 anisotropic XY model in a transverse field with periodic boundary conditions as the system under consideration <ref:2507.19994#pg3>. To account for the system-bath interaction, a full polaron transform is used to derive an effective Hamiltonian, denoted as H S, which is an accurate approximation of the mean force Gibbs state (HMF) <ref:2507.19994#pg3>. The effective Hamiltonian H S is given by Eq. (4), where the average of Cˆ calculated over the bath Hamiltonian is found as ⟨C⟩ˆ = exp-2Xk f 2k ν squared k coth(βνk 2) = exp 2g − 4g (βωc) squared ψ(1)(1 βωc)-1 <ref:2507.19994#pg4>. The effective Hamiltonian H S is rewritten as an anisotropic XY chain with parameters h, γ, and J <ref:2507.19994#pg5>.

Quantum Fisher Information (QFI) Derivation

The ultimate precision limit is quantified by the Quantum Fisher Information (QFI), which can be decomposed into classical and quantum contributions F(α) = F c + F q <ref:2507.19994#pg6>. The general expression for QFI for equilibrium states at WC is given by Eq. (9), while the SC case uses an analogous expression F(α) <ref:2507.19994#pg6>. The classical contribution to the QFI, F c, is related to the partition function and its derivatives via Eq. (D4) <ref:2507.19994#pg15>. The quantum contribution F q is derived by considering the parameter dependence of the eigenvectors via the Bogoliubov angle <ref:2507.19994#pg19>.

Finite-Size Effects and Phenomenological Approaches

Crucially, the work shows that neglecting finite-size (FS) effects leads to considerable errors in QFI calculations <ref:2507.19994#pg2>. The partition function Z given in Eq. (7) takes into account FS effects, and it is argued that in the limit N → ∞, the second and fourth term in Eq. (7), cancel and the first and the third term become equal <ref:2507.19994#pg4>. The QFI calculated with this assumption is dubbed as FPPA (F PPA) for both WC (SC) cases <ref:2507.19994#pg6>. Furthermore, a phenomenological QFI expression F'(α) is given by Eq. (11), which is derived using Hill’s nanothermodynamics <ref:2507.19994#pg5>. The results demonstrate that the phenomenological approach fails to accurately quantify precision limits because the effective free entropy and subdivision potential are independent of bath's and interaction Hamiltonian's parameters in some parameter domains <ref:2507.19994#pg6>.

Strong Coupling Effects on Precision

The study reveals that strong coupling (SC) can provide a slight metrological advantage for thermometry at small temperatures <ref:2507.19994#pg2>. SC reduces thermometry precision for higher temperatures but can improve it for lower temperature values <ref:2507.19994#pg2>. The maximum of QFI shifts to a higher value of h than the original phase transition point in the SC regime <ref:2507.19994#pg3>. For thermometry, F(β) shows an increase in precision at lower temperatures due to SC <ref:2507.19994#pg6>. The enhancement at higher β and g is attributed to environment-assisted transitions that repopulate thermally active states <ref:2507.19994#pg7>.

Scaling Behavior with System Size

The scaling behavior of the QFI with respect to N is investigated, and it is found that for larger values of N, a clear linear relationship emerges in the log-log plot <ref:2507.19994#pg11>. The results show that the scaling exponents are very close to the theoretical value of 1, indicating that in both cases the probe operates in the standard quantum limit <ref:2507.19994#pg11>. The FS effects play a non-negligible role in the scaling function away from the thermodynamic limit <ref:2507.19994#pg11>.

Conclusion

The research concludes that neglecting FS effects can cause a substantial error in calculating precision limits, but this error tends to get smaller with larger values of N and only in the thermodynamic limit does it vanish <ref:2507.19994#pg10>. The study suggests that for local coupling to a super-Ohmic bath, the phenomenological approach fails to give the correct QFI because it cannot describe the non-additive behavior of the spin chain system accurately <ref:2507.19994#pg16>.

How it works

The effective Hamiltonian H S is derived using a full polaron transform, which moves beyond perturbative methods <ref:2507.19994#pg3>. The effective Hamiltonian H S is an accurate approximation of the HMF, and its parameters are defined by h, γ, and J <ref:2507.19994#pg5>. This class of Hamiltonians can be diagonalized by Jordan-Wigner transformation followed by Fourier transformation and Bogoliubov-Valatin transformation <ref:2507.19994#pg5>.

Finite-Size Partition Function

The equilibrium state is written in its diagonal basis as ρˆeq = exp(-βHˆ) Z, where Z is the partition function that we intend to calculate <ref:2507.19994#pg13>. The total partition function is found by summing Z+ and Z−, which accounts for the contributions from positive and negative parity sectors <ref:2507.19994#pg14>. The full result for the partition function becomes Eq. (7) <ref:2507.19994#pg13>.

QFI Calculation Details

The QFI is calculated using the general expression F(α) = 4[⟨∂αψ∂αψ⟩ − ⟨∂αψψ⟩2] for a pure state <ref:2507.19994#pg14>. The classical contribution F c is related to the partition function and its derivatives via Eq. (D4) <ref:2507.19994#pg15>. The quantum contribution F q involves terms derived from the derivative overlap, which can be simplified to F q+ = 1/Z Xk∈K+ e−βϵk2 Z+ k (e−2βϵk − 1) squared e−2βϵk + 1 (∂θk/∂α) squared <ref:2507.19994#pg6>.

Subdivision Potential and Scaling

The subdivision potential E is a key thermodynamic variable in Hill’s framework, characterizing nanosystem’s non-additivity and vanishing in the thermodynamic limit <ref:2507.19994#pg17>. The phenomenological QFI is given as F'(α) = ∂2ψ'∂α2 + ∂2(βE)∂α2 <ref:2507.19994#pg17>. The microscopic subdivision potential E m is defined as E m = −β ∂H S/∂β + η <ref:2507.19994#pg18>. The results show that the microscopic subdivision potential E m and the phenomenological calculation of E using the spectrum of the effective Hamiltonian yield vastly different results <ref:2507.19994#pg27>.

Phase Transition Analysis

The phase transition point changes as h J → h J = ⟨C⟩ˆ h J squared (1 + γ) + ⟨C⟩ˆ 2(1 − γ). For larger values of g, for a fixed value of β considered in the β-h plane, the phase transition point shifts to a higher value of h. Therefore, for larger values of system-bath coupling, phase transition to the paramagnetic order occurs for higher magnetic field and lower temperatures.

Additional Results

The study provides results for QFI at SC calculated using microscopically derived effective Hamiltonian along with the phenomenologically calculated QFI <ref:2507.19994#pg13>. Comparing Fig. 12(b) with Fig. 12(c) and Fig. 12(e) with Fig. 12(f) shows that incorporating the spectrum of microscopically derived effective Hamiltonian into our phenomenological calculations forces the outcomes of QFI for both magnetometry and thermometry problems to largely agree at SC. The results for F(h) and F'(h) differ around the phase transition line <ref:2507.19994#pg29>.

Scaling Ansatz

The scaling function is found to be F(h) = NC∞(T, h) + C1(h, T) + C2(h, T) N + O(N −2) <ref:2507.19994#pg20>. The contribution to the QFI from this term is FR(T, h, N) = NB∞ R (T, h) + B1(h, T) + B2(h, T) N + O(N −2) <ref:2507.19994#pg20>. For zero temperature magnetometry, the ground state fidelity susceptibility diverges as F(h)/T = 0 ∼ N squared / ν = N squared <ref:2507.19994#pg20>. The scaling function for QFI with respect to the inverse temperature β away from criticality is F(β) = ND∞(T, h) + D1(h, T) + D2(h, T) N + O(N −2) <ref:2507.19994#pg20>.

Additional Results

The results for three different values of g are given in Fig.

1. Improved precision in metrology for spin chains by utilizing strong coupling (SC) effects for thermometry at low temperatures, as Our results reveal a potential advantage of SC for thermometry at low temperatures. This allows AI systems designed for quantum sensing to achieve enhanced precision when operating near the critical field point where SC effects are maximized.

2. Development of a robust QFI calculation method that explicitly incorporates finite-size (FS) effects, as the paper shows neglecting FS effects leads to considerable errors in QFI calculations. This enables AI models to accurately predict measurement sensitivity limits for small systems away from the thermodynamic limit, preventing overestimation or underestimation of precision.

3. Creation of a hybrid metrological model combining microscopic and phenomenological approaches that correctly captures non-additive thermodynamic behavior, as the paper demonstrates the phenomenological approach is incapable of quantifying the precision limits accurately when neglecting FS effects. This allows AI systems to select the appropriate theoretical framework based on system size and coupling regime for optimal performance.

4. Implementation of a control strategy for magnetometry where the anisotropy parameter can be tuned to enhance precision, as demonstrate enhanced magnetometric precision through control of the anisotropy parameter. This enables AI-driven experimental setups to dynamically optimize system parameters to maximize sensitivity in specific magnetic field ranges.

5. Enhanced understanding and prediction of phase transition behavior by modeling how the system-bath coupling affects the critical point, as SC with the local bath causes the phase transition point to change as hJ→hJ. This allows AI systems to predict operational shifts in quantum devices when environmental coupling strength is varied, guiding experimental design towards higher magnetic fields and lower temperatures for specific orders.

6. Improved scaling analysis of QFI with respect to system size (N) using the derived ansatz Fmax(α) = aNξ + b/N + c, which shows that for larger values of N a clear linear relationship emerges. This allows AI systems to accurately predict the asymptotic performance of quantum probes as system size increases, confirming operation in the standard quantum limit and identifying scaling exponents.

7. Optimization of thermometry protocols by exploiting environment-assisted transitions, as increasing g boosts F(β) for large values of β (small temperature values T). This enables AI to design operational sequences that target lower temperatures more effectively by increasing system-bath coupling strength (g), shifting the peak sensitivity to desired low-temperature regimes.

8. Verification of microscopic vs. phenomenological QFI calculations by comparing results, as incorporating the spectrum of microscopically derived effective Hamiltonian into our phenomenological calculations forces the outcomes of QFI for both magnetometry and thermometry problems to largely agree at SC. This provides a rigorous validation layer for AI-derived models, ensuring that simplified models accurately reflect complex microscopic dynamics.

Equilibrium quantum metrology is often formulated under assumptions of weak system-bath coupling and thermodynamic limit scaling, which may not be valid for realistic finite-size (FS) quantum devices. Here, we establish how strong coupling (SC) and FS effects jointly modify achievable precision bounds for equilibrium quantum probes. Considering a transverse-field anisotropic XY spin chain coupled to a thermal environment, first utilize we derive a full polaron transform to derive the effective Hamiltonian at equilibrium. Then, using the bare Hamiltonian and the obtained effective Hamiltonian and taking into account the role of FS effects in the calculation of partition function, we derive an analytic expressions for the quantum Fisher information (QFI) for a general parameter for the equilibrium state in weak coupling (WC) and SC regimes, respectively. In contrast to this microscopic approach, we also utilize the Hill's phenomenological nanothermodynamics to calculate an effective QFI expression at SC. Our results, presented for specific cases of magnetometry and thermometry, reveal that considering FS and SC effects lead to substantial and systematic deviations in QFI, even for moderate system sizes, and that commonly used thermodynamic-limit approximations can significantly misestimate precision. This work also highlights the inadequacy of phenomenological approaches in describing the metrological capability and thermodynamic behavior of systems at SC. Our work establishes a consistent framework for evaluating metrological performance of realistic, finite quantum devices operating beyond the weak-coupling paradigm.