Precise measurement of physical parameters requires sufficient resources (N). Classically, the standard quantum limit (SQL) restricts sensitivity to 1/√N. Quantum resources, however, allow surpassing this limit, potentially reaching the Heisenberg limit (HL) of π/N. Achieving the HL demands efficient resource allocation, especially in ab-initio parameter estimation (without prior knowledge). Independent resource use results in SQL scaling; optimal sensitivity leverages quantum correlations during probe preparation. NOON states, maximally entangled two-mode states, have shown promise, but experiments are limited to small N due to generation complexity and sensitivity to losses. Alternative entanglement-free multi-pass strategies, though achieving Heisenberg scaling, suffer exponential loss sensitivity. These limitations hinder the observation of Heisenberg-limited performance for large N. This paper focuses on the non-asymptotic regime, relevant for practical quantum sensors, developing strategies to achieve N⁻¹ scaling for finite N (the non-asymptotic Heisenberg limit). The authors employ NOON-like states encoded in total photon angular momentum (more robust to losses), implementing a method for optimal resource allocation. This approach requires high-dimensional NOON-like superposition states with tunable dimensionality and individual resource allocation at the single-photon level. The protocol is non-adaptive, requiring only offline pre-calculated procedures, and is tested experimentally for an ab-initio measurement of an unknown rotation angle within [0, π). This method resolves ambiguities among equivalent angle values, demonstrating sub-SQL performance across a wide N range.

Several approaches have been explored for achieving Heisenberg scaling in parameter estimation. NOON states, a class of maximally entangled states, have been widely used in photonic platforms for phase estimation, particularly in interferometric measurements. However, scaling NOON states to large N is challenging due to their non-deterministic generation with linear optics and high sensitivity to losses. Alternative methods involve entanglement-free, multi-pass strategies for ab-initio phase estimation. While these methods also achieve Heisenberg scaling, they remain restricted to small N due to their sensitivity to losses when increasing the number of passes. The literature highlights the challenges of maintaining Heisenberg scaling with increasing resources and the detrimental effect of losses in both multi-particle entangled states and multi-pass protocols. These lossy scenarios typically result in an overall efficiency scaling as N_overall ~ n⁻¹, where n represents the number of resources. Optimized states can mitigate this, but only to a constant factor improvement, emphasizing the significance of the non-asymptotic regime in practical quantum sensing.

The proposed method aims to recover an unknown parameter θ ∈ [0, π) (represented by an optical phase shift or rotation angle). The approach uses *n* copies of the input state (|0⟩ + |1⟩)/√2, transforming each into |ψ_s(θ)> = (|0⟩ + e^-i2sθ|1⟩)/√2. The integer *s* represents the quantum resources allocated to each copy. The total number of operations is *ns*. The authors utilize non-entangled resources in a coherent superposition of different total angular momentum *s* states. The method employs a multistage procedure with increasing quantum resources s₁, s₂, ..., sₖ, passing information stage-by-stage to disambiguate θ as sensitivity grows. The s values are pre-calculated, not adaptively chosen. At each stage, *nᵢ* copies of |Ψᵢ(θ)| are measured non-adaptively, constructing an ambiguous multivalued estimator. Plausible intervals for θ are identified, and one interval is deterministically chosen based on previous stage information. The number of copies *nᵢ* is optimized to avoid fringe ambiguity. The algorithm may incur errors at each stage, but the error probability decreases with increasing *nᵢ*. The precision of the final estimator is optimized over the number of probes n₁, n₂, ..., nₖ, while keeping the total resources N = Σᵢ nᵢsᵢ constant. The protocol is non-adaptive, decoupling measurement and algorithmic data processing. The experiment employs the total angular momentum of single photons to measure the rotation angle θ between two reference frames. The setup uses q-plates (QPs), which modify photons' OAM based on their polarization. Single photon pairs are generated, and one photon acts as a trigger while the other (signal) propagates through the apparatus. The probe state is prepared by initializing the polarization, passing through a QP and a half-wave plate (HWP) to create a superposition. The rotated measurement station encodes the rotation in the photon state via a phase shift, and a second QP and HWP reconvert it into a polarization state for measurement with a PBS and single-photon detectors. The interference fringes oscillate with probability P = cos²[(m+1)θ], where m is the OAM, providing unambiguous estimation of rotations in [0, π). The estimation error is Δθ ≥ 1/(2(m+1)√n). The apparatus is fully automated, with six QPs in a cascaded configuration (three in preparation, three in measurement), mounted in a motorized rotation stage. The experiment uses four OAM configurations (s = 1, 2, 11, 51). The circular error is calculated as θ̃ − θ = −(θ̃ − θ) mod π, and the root-mean-square error (RMSE) is averaged over 17 different rotation angles. The data processing algorithm involves calculating estimators at each stage, identifying plausible intervals, and selecting one based on previous intervals. Optimization involves minimizing an upper bound on the RMSE while fixing the total resources N. A multistage strategy is used, adding stages progressively as N increases. Numerical simulations help determine the best set of quantum resources to use.

The experiment demonstrates sub-SQL performance for a wide range of resources N (O(30,000)). Using single-photon states with progressively higher-order total angular momentum achieves this, progressively approaching Heisenberg scaling. An error reduction of up to 10.7 dB below the SQL is achieved. Global analysis fitting the uncertainty with C/N^α reveals that α is compatible with the SQL (α = 0.5) when only s = 1 states are used. However, sub-SQL performance is obtained with s > 1, with α reaching 0.7910 ± 0.0002 for 6460 resources and 0.6786 ± 0.0001 for 30,000 resources. A local analysis focusing on regions defined by the maximum s value used confirms sub-SQL scaling and even regions compatible with Heisenberg scaling (α = 1) in significant resource ranges (O(1,300)). These results show the effectiveness of employing higher-order OAM states to enhance precision, and the method works even with s values different from the theoretically optimal ones, indicating its versatility.

The results demonstrate Heisenberg-like precision for a broad range of resources N, exceeding previous photonic experiments. The non-adaptive protocol using single-photon states with high total angular momentum in a fully automated setup significantly improves upon state-of-the-art sub-SQL estimation precision for large N. Global and local analyses confirm the sub-SQL and near-Heisenberg scaling. The flexibility of the method, even when deviating from theoretically optimal s values, highlights its practical applicability. This approach offers a robust and versatile method for optimizing resource utilization in ab-initio parameter estimation, with potential applications in sensing, quantum communication, and information processing.

This work experimentally demonstrates a protocol for achieving sub-SQL performance in parameter estimation across a wide resource range. Using high-order OAM single-photon states and a non-adaptive approach, the experiment achieves significant improvement over existing methods. The results highlight the practical potential of this method for various applications, especially those requiring robust and efficient resource utilization. Future work could explore further optimization strategies, investigating different state choices and extending the range of accessible OAM values.

While the experiment demonstrates significant improvement in achieving Heisenberg-like scaling for a wide resource range, limitations exist. The experiment relies on pre-calculated resource allocation strategies. The accuracy of the results depends on the accuracy of the theoretical model and the assumptions made. The experimental setup might have inherent limitations that could affect the achievable precision. Furthermore, the use of single photons with high OAM may introduce challenges in terms of generation, propagation, and detection efficiency.