Single-shot measurement learning as a self-certifying estimator for quantum-enhanced sensing

Abstract

Single-shot measurement learning (SSML) learns a compensation unitary from a one-bit success/failure record and halts after a prescribed run of consecutive successes. We recast SSML as an adaptive estimator on a parameterized sensing manifold and ask what role it can play in quantum-enhanced sensing. First, we show that the terminal run itself furnishes an intrinsic certificate of local alignment: longer terminal runs certify smaller infidelity, and near the optimum this becomes a Fisher-calibrated certificate of parameter error. Second, for compensation-type sensing families, the Bernoulli success/failure record is locally matched to the probe quantum Fisher information (QFI), so SSML preserves the probe's metrological content despite using only one classical bit per copy. In this sense, SSML makes the quantum enhancement carried by the probe operationally available in an online self-terminating protocol. Applied to GHZ/NOON probes of depth $m$, SSML retains the familiar square-root entanglement gain over product probes at fixed total resource, while an ideal multiscale architecture remains compatible with Heisenberg scaling. Monte Carlo simulations of photonic NOON-state phase sensing show the expected near-inverse decay of terminal infidelity with entangled shots, SQL-like total-resource scaling at fixed entanglement depth, the corresponding fixed-resource entanglement gain, the global limitation of a single fringe scale, and the recovery of Heisenberg-compatible behavior under ideal multiscale hand-off. These results identify SSML as a Fisher-preserving, self-certifying estimator layer for quantum-enhanced sensing.

Figures (5)

• Figure 1: Schematic: SSML as an adaptive sensing loop. The unknown sensing map $\hat{V}_\lambda$ acts on a probe family, a compensation $\hat{U}(\tilde{\lambda})$ is learned shot by shot, and a yes/no projection onto the fiducial probe feeds a one-bit record back to the controller. The same protocol naturally separates into acquisition/search and lock/certification stages.
• Figure 2: Certificate and local geometry. (a) The terminal run $M_H$ induces the intrinsic certificate scale $\epsilon_{\rm cert}=1-\eta^{1/M_H}$, with the expected $\ln(1/\eta)/M_H$ asymptote. (b) Near the optimum, the infidelity of a compensation family is locally quadratic, $1-p_s \approx x^2/4$ with $x=\sqrt{F_Q}(\tilde{\lambda}-\lambda)$, i.e., the standard QFI/Bures metric relation.
• Figure 3: Numerical illustration for photonic NOON-state SSML. (a) Mean terminal infidelity versus entangled shots $\nu=\mathbb{E}[T]$ for $m=1,2,4,8$ in the local branch-resolved regime. The fit gives $\mathbb{E}[\epsilon_T]\propto \nu^{-0.95}$. (b) Phase RMSE versus total photon number $R=m\,\mathbb{E}[T]$. The slope is SQL-like, $\Delta\theta\propto R^{-0.48}$, while the entangled curves are shifted downward. (c) At fixed total resource, the prefactor obeys $\sqrt{R}\,\Delta\theta\propto m^{-1/2}$, confirming the preserved $\sqrt{m}$ advantage.
• Figure 4: Single-scale aliasing under a global prior. (a) For a global prior and fixed $m=8$, the terminal infidelity continues to decrease with the total photon number. (b) The global phase RMSE, however, saturates because the protocol can lock onto an incorrect fringe. This is the metrological reason that the local fixed-depth numerical illustration of Fig. \ref{['fig:noon-main']} is deliberately restricted to a branch-resolved regime.
• Figure 5: Ideal multiscale NOON-SSML. A coarse-to-fine architecture with stage depths $m_j=2^j$ and ideal branch-resolved hand-off exhibits the Heisenberg-compatible trend $\Delta\theta_{\rm final}\propto R_{\rm tot}^{-0.99}$ in the noiseless regime. The figure should be read as a resource-counting illustration rather than as a complete branch-selection protocol.

Theorems & Definitions (7)

• Proposition 1: Fisher-calibrated parameter certificate
• proof : Proof sketch
• Theorem 1: Local Fisher matching of SSML
• proof : Proof sketch
• Corollary 1: Quantum gain at fixed entanglement depth
• proof : Proof sketch
• Proposition 2: Ideal multiscale Heisenberg compatibility