Dicke superposition probes for noise-resilient Heisenberg and super-Heisenberg Metrology

TL;DR

This work addresses phase estimation in noisy quantum metrology using Dicke-state superpositions under both linear ($H_1=\hat{\mathbf{J}}_{\boldsymbol{n}}$) and nonlinear two-body encodings ($\hat{H}^{(r)}_2$). It derives explicit QFI expressions, showing near-Heisenberg scaling for selective Dicke superpositions under linear encoding and establishing nonlinear Heisenberg scaling for four representative two-body Hamiltonians via optimal and near-optimal probes. The study analyzes robustness against phase damping, amplitude damping, and global depolarization, finding that near-optimal Dicke superpositions often rival or surpass truly optimal probes in noisy settings, and can approach nonlinear HL benchmarks in several regimes. These results position Dicke-state superpositions as versatile, experimentally accessible resources for high-precision, noise-resilient quantum metrology across linear and two-body interaction paradigms.

Abstract

Phase sensing with entangled multiqubit states in the presence of noise is a central theme of modern quantum metrology. The present work investigates Dicke state superposition probes for quantum phase sensing under parameter encoding generated by one- and two-body interaction Hamiltonians. A class of N-qubit Dicke superposition states that exhibit near-Heisenberg scaling, of the quantum Fisher information, while maintaining significantly enhanced robustness to dephasing noise compared to GHZ, W-superposition, and balanced Dicke states, under unitary encodings generated by one-body interaction Hamiltonians are identified. For two-body interactions, Dicke superposition probes optimizing the quantum Fisher information are identified, and their performance under phase-damping, amplitude-damping, and global depolarizing noise is explored. Within this family, certain Dicke superpositions are found to combine super-Heisenberg scaling with improved resilience to phase damping relative to Fisher information optimal probes. These results establish tailored near-optimal Dicke-state superposition probes as versatile and noise-resilient resources for Heisenberg and super-Heisenberg quantum phase sensing governed by one- and two-body interactions.

Figures (7)

• Figure 1: QFI under collective spin encoding $\hat{J}_{\boldsymbol n}$ for near-optimal Dicke superposition states $\lvert D^{(N)}_{l,l'}\rangle$, compared with the W--superposition state $\lvert D^{(N)}_{1,N-1}\rangle$, the balanced Dicke state $\lvert D_{N/2,N/2}\rangle$ (for even $N$), and the GHZ state. The Dicke superposition probes exhibit a modest but systematic enhancement in QFI relative to the W--superposition state over an intermediate regime $N\lesssim 20$.
• Figure 2: QFI for $N=8$ probe states as a function of the noise parameter $p$ under collective-spin encoding Hamiltonian $\hat{H}_1=\hat{J}_{\boldsymbol{n}}$. The near-optimal Dicke superposition state $\lvert D^{(8)}_{3,5}\rangle$ is compared with the GHZ, W--superposition $\lvert D^{(8)}_{1,7}\rangle$, and balanced Dicke state $\lvert D_{4,4}\rangle$ under (a) phase damping and (b) amplitude damping and (c) global depolarization.
• Figure 3: Phase sensitivity $\Delta\theta$ of $N=8$ probe states as a function of the noise parameter $p$ under encoding Hamiltonian $\hat{H}_1=\hat{J}_{\boldsymbol{n}}$. The near-optimal Dicke superposition state $\lvert D^{(8)}_{3,5}\rangle$, the GHZ state, the W--superposition state $\lvert D^{(8)}_{1,7}\rangle$, and the balanced Dicke state $\lvert D_{4,4}\rangle$ are compared under (a) phase damping, (b) amplitude damping, and (c) global depolarization. The HL and the SNL are indicated for reference.
• Figure 4: Comparison of phase sensitivity $\Delta \theta$ of $N=8$-qubit optimal probe $\lvert \psi_{(N=8)}^{(2)}\rangle_{\boldsymbol{n}}=\dfrac{1}{\sqrt{2}} \left(\lvert 0\rangle_{\boldsymbol n}^{\otimes 8}+\lvert D^{(8)}_{4,5}\rangle_{\boldsymbol n}\right)$ (see (\ref{['psiopt']})) with parameter encoding generated by the two-body interaction Hamiltonian $\hat{H}^{(2)}_2=\hat{J}_{\boldsymbol{n}} + \hat{J}_{\boldsymbol{n}}^2$. The plots indicate the relative robustness of the optimal probe under phase damping, amplitude damping, and global depolarization. The NL-SNL and the NL-HL are also indicated.
• Figure 5: Comparison of phase sensitivity $\Delta \theta$ of $N=8$-qubit optimal probe $\lvert \psi_{(N=8)}^{(2)}\rangle_{\boldsymbol{n}}=\dfrac{1}{\sqrt{2}} \left(\lvert {\rm GHZ}\rangle_{\boldsymbol n}+\lvert D_{4,4}\rangle_{\boldsymbol n}\right)$ with that of near optimal Dicke superposition probe $\vert D^{(8)}_{0,2}\rangle_{\boldsymbol{n}}$, both encoded via the two-body interaction Hamiltonian $\hat{H}^{(1)}_{2}=\hat{J}^2_{\boldsymbol{n}}$, subjected to global depolarization. The NL-SNL and the NL-HL are displayed for reference.