Experimental metrology beyond the standard quantum limit for a wide resources range

The study addresses how to estimate a physical parameter (here, a rotation angle θ ∈ [0, π)) with precision that surpasses the standard quantum limit (SQL), which scales as 1/√N when only classical resources are used. Quantum strategies can in principle achieve Heisenberg-limit (HL) scaling, π/N, by employing N quantum resources. However, practically attaining HL scaling across a broad range of resources is challenging due to state preparation complexity, loss sensitivity, and ambiguity issues in ab-initio estimation without prior knowledge. The paper investigates a non-asymptotic regime where quantum advantages are most relevant for real sensors with limited resources, and proposes a resource allocation strategy that achieves sub-SQL scaling and locally follows the Heisenberg 1/N power law, without adaptive measurements or prior parameter information. The target application is unambiguous estimation of rotation angles between platforms, relevant for spatial synchronization in communication systems.

Prior approaches include NOON states enabling super-resolving interference and HL sensitivity in interferometry, with applications in imaging and biosensing. However, NOON states are hard to scale (deterministic generation with linear optics not possible for N>2), with experiments up to 10 photons and strong loss sensitivity that quickly erodes quantum advantage; unconditional sub-SQL precision accounting for total resources has only recently been shown with two-photon states. Entanglement-free ab-initio phase estimation via multi-pass strategies (adaptive and non-adaptive) can achieve HL scaling but becomes exponentially sensitive to losses and is thus confined to small N. In general, in noisy scenarios the asymptotic advantage reduces to at best a constant factor, and protocols must be optimized for non-asymptotic regimes. High-dimensional photonic orbital angular momentum (OAM) offers large resource values per photon and greater robustness to loss compared to multi-particle NOON or multi-pass strategies. The present work builds on these insights to extend the range over which sub-SQL and HL-like scaling can be observed.

Estimation protocol: The parameter θ ∈ [0, π) is encoded into states |ψ_s(θ)⟩ = (|0⟩ + e^{-i 2 s θ}|1⟩)/√2, where s denotes the quantum resource allocated per probe. A multistage, non-adaptive procedure uses stages i=1..K with precomputed resource values s_i and numbers of copies n_i (total resource N = Σ_i n_i s_i). Each stage yields an ambiguous estimator due to periodicity π/s_i; the algorithm constructs s_i candidate intervals and disambiguates by selecting the unique interval overlapping the one chosen at the previous stage. Resource allocation n_i is optimized (offline) to minimize the final estimator variance subject to fixed N. It is analytically known that choices like s_i=2^{i-1} can yield Heisenberg scaling with appropriate n_i distributions. Non-unit visibilities and losses are incorporated in the optimization; measurement data are post-processed to produce the estimate without adaptive updates during acquisition. Experimental platform: Single-photon pairs at 808 nm are generated via degenerate type-II SPDC in a 20-mm ppKTP crystal pumped by a 404 nm cw laser. The idler photon triggers detection; the signal photon is prepared in |H⟩ polarization, then passes a q-plate (QP) of charge q and a half-wave plate to produce a vector vortex superposition |Ψ0⟩ = (1/√2)(|R⟩|m⟩ + |L⟩|−m⟩), with m=2q. Including spin and orbital contributions, the total angular momentum in the two components is ±|m+1|, and s=m+1 serves as the per-photon resource. A relative rotation θ between preparation and measurement stations imprints a phase 2|m+1|θ between the components: |Ψ1⟩ = (1/√2)(e^{i(m+1)θ}|R⟩|m⟩ + e^{-i(m+1)θ}|L⟩|−m⟩). A second HWP and a QP with the same q (aligned with the rotated station) convert the state to polarization: |Ψ2⟩ = (1/√2)(|R⟩ + e^{-i 2(m+1)θ}|L⟩), which is measured by a PBS and APDs. The transmission probability oscillates as P=cos^2[(m+1)θ], with periodicity π/(m+1). Measurements split evenly between H/V and D/A bases; the station is motorized to control θ. Resource settings: Six QPs are mounted in a cascaded, motorized rotation stage; during acquisition, only a matched QP pair (one in preparation, one in measurement) is active per step. The accessible resource values are s=1 (all QPs off), s=2, 11, 51, corresponding to q=0, 1/2, 5, 25 with the relation s=2q+1. Precision bounds and SQL/HL relations: For n photons at resource s=m+1, the estimation error satisfies Δθ ≥ 1/[2 (m+1) √n], showing Heisenberg-like dependence on s and SQL dependence on n. If s is fixed and n→∞, scaling returns to SQL in total resources. The protocol targets non-asymptotic regimes by increasing s across stages to follow HL-like 1/N behavior. Data processing and optimization: For each stage, from frequencies f_HV and f_DA, an estimator φ̂_i = atan2(2 f_HV,i − 1, 2 f_DA,i − 1) ∈ [0, 2π) is formed (equivalent to θ after scaling). This yields s_i candidate intervals centered at φ̂_i + 2π m / s_i; the algorithm selects the unique interval overlapping the one from the prior stage. Interval sizes are set via a recursive γ_i to ensure unique overlap. Error probabilities are bounded via Hoeffding-inspired bounds, leading to an upper bound on the RMSE of the final estimator, which is minimized by Lagrangian optimization over n_i given fixed N and chosen s_i. Non-unit visibility v_i and loss η_i are included by scaling parameters in the bound (for post-selected experiments η≈1, visibility dominates). Numerical simulations and analytical bounds guide when to introduce higher-s stages as N grows, potentially using a subset of available s_i for optimal performance. Acquisition and analysis: For each strategy (sets of s used), measurements are repeated R=200 times across 17 θ values in [0, π); circular distance is used to compute RMSE per strategy and averaged over θ. Fits of Δθ versus N to C/N^α quantify scaling exponents globally and locally within N regions dominated by a given maximum s.

- Demonstrated sub-SQL precision over a wide resource range N up to O(30,000) total resources using single photons with high total angular momentum (OAM-based NOON-like states). - Achieved variance reduction exceeding 10 dB below the SQL (up to 10.7 dB) when employing the full set s = 1, 2, 11, 51 with optimized n_i. - Global scaling analysis (fit Δθ ≈ C/N^α): • Using all QPs (s = 1, 2, 11, 51), maximum α = 0.7910 ± 0.0002 at N = 6460. • Over the full dataset up to N ≈ 30,000, α = 0.6786 ± 0.0001, remaining clearly above the SQL exponent 0.5. • Considering only N ≥ 62 (first inclusion of s > 1), α increases to 0.8301 ± 0.0003 at N = 4764. - Local scaling analysis with s = 1; 2; 11; 51: • For s = 1 only (2 ≤ N ≤ 60), α ≈ 0.5 (compatible with SQL). • Introducing s = 2 (62 ≤ N ≤ 264) yields sub-SQL scaling (0.5 < α ≤ 0.75). • With s up to 11 and 51 (N > 264), α > 0.75, approaching the Heisenberg 1/N power law. • Identified two N intervals (266 ≤ N ≤ 554 and 1772 ≤ N ≤ 2996) where α is compatible with 1 (within 3σ), i.e., locally following the Heisenberg scaling over ranges of ~300 and ~1000 resources, respectively. - Verified that if s is held fixed beyond some N, scaling reverts toward SQL, highlighting the need to increase s to maintain HL-like behavior. - The protocol resolves ambiguity over the full [0, π) range without adaptive measurements and tolerates non-unit visibility through pre-optimized resource allocation.

The work addresses the challenge of maintaining Heisenberg-like precision over a broad, practically relevant range of total resources. By encoding resources in the total angular momentum per photon and distributing measurements across non-adaptive, precomputed stages with increasing s, the protocol disambiguates periodic estimates and concentrates precision where it is most effective. Compared to multiphoton NOON states and multipass protocols, the OAM-based single-photon approach is more resilient to loss and avoids exponential fragility, enabling sub-SQL scaling over tens of thousands of total resources. The global and local scaling analyses quantitatively confirm extended regions where the precision follows the same 1/N power law as the Heisenberg limit, showing that careful offline resource allocation can deliver quantum-inspired enhancements in non-asymptotic regimes relevant to actual sensors. This capability is directly applicable to robust estimation of rotation angles, such as aligning reference frames in quantum communication systems, and suggests a general strategy for optimizing quantum resources in a variety of sensing contexts.

The paper demonstrates an experimental, non-adaptive protocol that achieves sustained sub-SQL precision and locally Heisenberg-like scaling over extended resource ranges by optimally allocating per-photon total angular momentum resources and sample sizes across multiple stages. Using single-photon OAM states up to s=51, the experiment achieves >10 dB variance reduction below SQL and identifies broad N intervals with scaling compatible with 1/N. The approach operates with offline optimization, is robust to non-unit visibility, and provides a versatile toolbox for ab-initio parameter estimation without prior information. Future directions include extending to higher s values and additional resource stages, integrating loss-aware, non-postselected operation to quantify unconditional advantages, applying the method to other parameters and platforms (e.g., different degrees of freedom or matter systems), and deploying in practical quantum sensing and communication scenarios requiring precise rotational or phase alignment.

- Advantage relies on increasing s; with s fixed and N growing, scaling reverts toward SQL, limiting asymptotic performance without access to higher resources. - The demonstrated resource set is limited (s = 1, 2, 11, 51); broader or denser s sequences could further extend HL-like regions but require more complex generation and stable handling of high-OAM modes. - Non-unitary conversion efficiency and reduced visibility of q-plates impact precision; although incorporated in optimization, they cap achievable performance. - Photon loss reduces detected counts; parts of the analysis consider post-selection (effectively η≈1), which may overestimate performance compared with fully unconditional, loss-inclusive benchmarks. - The Heisenberg-like behavior is verified in non-asymptotic, finite-N regions; truly asymptotic performance in noisy settings generally degrades to constant-factor improvements. - Implementation complexity grows with higher s due to beam divergence, alignment, and detection challenges for high-order OAM states.