Distributed quantum sensing of multiple phases with fewer photons

The study addresses how to achieve quantum-enhanced sensitivity for estimating linear combinations of multiple spatially distributed phases when entangled probe photon numbers are fewer than the number of parameters (nodes). Classical strategies are bounded by the standard quantum limit (SQL), while distributed quantum sensing aims to surpass this by leveraging entanglement across distant sensors. Prior discrete-variable approaches required multi-photon, multi-party entangled states with photon number at least equal to the number of phases, limiting scalability due to the difficulty of generating large-photon-number entanglement. The paper’s purpose is to propose and demonstrate a protocol that enables quantum enhancement and Heisenberg scaling even when the total photon number N is less than the number of unknown phases d, thereby improving practicality for large sensor networks.

Previous work in continuous-variable (CV) and discrete-variable (DV) quantum metrology established that entanglement can yield sensitivities beyond separable strategies. CV networks have demonstrated improvements using multipartite entanglement and error correction. DV approaches using mode- and particle-entangled (MePe) states can, in principle, achieve Heisenberg scaling, and experiments have estimated averages of two or three phases. However, these DV strategies required the number of photons to be at least equal to the number of parameters (e.g., two photons for two phases, six photons for three phases), presenting a significant scalability bottleneck because generating high-photon-number entanglement is challenging. These constraints motivate the need for protocols that reduce photon requirements while retaining quantum advantage.

Theory: Consider d unknown phases φ = (φ1, …, φd) distributed across nodes, and the goal is to estimate a linear global function φ̄ = α^T φ with Σi αi = 1. The probe state |Ψ⟩ evolves under U(φ) = exp(−i Σj Hj φj) and is measured with projectors {Πi}, yielding probabilities {Pi}. Estimation is performed via maximum likelihood estimation (MLE). The uncertainty obeys a Cramér–Rao bound based on the Fisher information matrix. Traditional MePe states require N ≥ d. The authors propose a different N-photon two-mode polarization-entangled superposition state that distributes N/2 photons across adjacent node pairs around a ring, enabling probing of multiple paths via a beam splitter network (BSN). This construction only requires that N be even and allows estimation of d phases even for N < d. Generalization shows a Fisher information matrix with nonzero diagonal and nearest-neighbor elements, yielding a bound Δφ̄^2 = 1/N^2 (Heisenberg scaling), equivalently 1/(n^2 d^2) with n = N/d. Experimental setup: A two-photon (N=2) polarization-entangled Bell state |φ+⟩ = (|HA HB⟩ + |VA VB⟩)/√2 is generated via SPDC using a 10-mm type-II PPKTP crystal in a Sagnac interferometer. The state is multiplexed by a BSN of two 50/50 fiber beam splitters to distribute entanglement among four nodes (d=4) arranged as adjacent pairs (12, 23, 34, 41). Each node is 3 km of fiber away from the central source. Phase encoding at each node uses a QWP–HWP–QWP sequence. Measurements perform σz projections via an HWP at 22.5° and a PBS, with coincidences detected on SNSPDs. The resulting two-photon coincidence probabilities for node pairs jk ∈ {12, 23, 34, 41} follow: Pjk++ = Pjk−− = [1 + V cos(Φj + Φk)]/16 and Pjk+− = Pjk−+ = [1 − V cos(Φj + Φk)]/16, where V is the visibility. Data acquisition: Interference fringes are recorded by scanning phases; for sensitivity estimation, Φ1 is scanned with Φ2 = π/2 and Φ3 ≈ Φ4 ≈ 0 fixed. Sixteen outcome probabilities are estimated from post-selected coincidence events, with μ ≈ 367 measurement samples used in MLE to estimate the average phase Φ = −(1/4) Σj φj and its standard deviation. Expected limits (weak CRB), SQL (1/√(μN)), and HS (1/(√μ N)) are computed using measured visibilities.

- Demonstrated distributed estimation of the average of four spatially separated phases using only two entangled photons (N=2, d=4), with each node 3 km from the central source. - Achieved a 2.2 dB sensitivity improvement over the standard quantum limit (SQL); standard deviations approach the Heisenberg scaling (HS) bound. - Measured average visibilities for selected pairwise outcome probabilities P++: 0.955 (12), 0.981 (23), 0.970 (34), and 0.945 (41). - Utilized μ ≈ 367 post-selected coincidence samples for MLE-based phase estimation across sixteen outcome probabilities. - Theoretical analysis shows the Fisher information structure of the proposed state yields Δφ̄^2 = 1/N^2, implying HS with respect to total photon number, and equivalently Δφ̄^2 = 1/(n^2 d^2) with n = N/d; notably, the protocol enables quantum enhancement even when N < d.

The work addresses the core challenge of scaling distributed quantum sensing by eliminating the constraint that the photon number must be at least the number of unknown phases. By employing a superposition of two-mode entangled states distributed across adjacent node pairs and multiplexed via a beam splitter network, the protocol maintains quantum correlations sufficient to achieve Heisenberg-limited scaling in principle. Experimentally, despite non-idealities (fiber-induced phase fluctuations and finite sampling), the results surpass the SQL by 2.2 dB and approach HS, validating the concept with practical entangled photon sources and standard components over kilometer-scale fibers. The findings signify that large-scale sensor networks can leverage fewer quantum resources while retaining strong metrological advantages, and that further improvements in stability, visibility, and detection efficiency can push performance to the theoretical limit.

The paper introduces and experimentally validates a distributed quantum sensing protocol that estimates multiple phases with fewer photons than parameters, demonstrating a 2.2 dB enhancement over the SQL in estimating the average of four phases using a two-photon entangled state over 3 km links. Theoretically, the approach achieves Heisenberg scaling Δφ̄^2 = 1/N^2 without requiring N ≥ d, and generalizes to arbitrary even N and number of nodes d. Future directions include increasing node counts for larger networks, improving visibility and stability, employing high-efficiency photon-number-resolving detectors, leveraging multi-pass strategies, deploying over real fiber networks, and integrating with simultaneous multiparameter estimation at single locations.

- Post-selection was used; losses and detector inefficiencies (lack of high-efficiency photon-number resolving detectors) were not included in the primary sensitivity analysis. - Phase fluctuations over kilometer-scale fibers (~3 km) and limited sample size (μ ≈ 367) led to mismatches between measured data and ideal probability models; this contributed to error bars and, in some cases, apparent deviations relative to HS bounds. - Non-ideal visibilities limited performance; improvements in stability and detection could bridge the gap to the Heisenberg limit. A loss-inclusive analysis is provided in the Supplementary Note.