Quantum metrology offers enhanced sensitivity in parameter estimation, exceeding the standard quantum limit (SQL) achievable through classical means. A significant advancement is distributed quantum sensing, which aims to estimate linear combinations of multiple unknown parameters spread across distant nodes, unlike conventional methods focusing on single locations. This capability is valuable for applications such as local beam tracking and global clock synchronization. While various strategies using entangled probe states among distributed sensors have been proposed for distributed multiple-phase sensing, achieving quantum-enhanced sensitivity often necessitates entangled resources with a photon number equal to or greater than the number of unknown phases. This requirement poses a significant challenge as the number of nodes increases, limiting scalability. This research introduces a new protocol that overcomes this limitation, enabling quantum-enhanced sensitivity even when the number of photons is less than the number of unknown phases.

Continuous-variable (CV) quantum metrology has shown that entangled CV states can improve sensitivity compared to separable states. In discrete-variable (DV) quantum metrology, theoretical work has established that Heisenberg scaling (HS) is achievable using mode-entangled and particle-entangled (MePe) states, with experimental demonstrations focusing on estimating the average of a small number of phases. However, existing methods using MePe states necessitate a photon number equal to or greater than the number of unknown phases. Generating these multi-photon entangled states presents a major obstacle to scalability in large-scale distributed quantum sensing.

This study proposes a novel distributed quantum sensing protocol that achieves quantum-enhanced sensitivity even when the photon number is fewer than the unknown phases. The researchers experimentally demonstrate the estimation of the average of four spatially separated phases (distributed 3km away from the central node) using a two-photon entangled state. The protocol involves distributing a prepared two-photon entangled state across four nodes. Each node incorporates phase encoding, and local measurements are performed at each node. The average of the four unknown phases is estimated using the maximum likelihood estimator (MLE). The experimental setup includes a Sagnac-interferometer-based Bell source generating a polarization Bell state, which is then split using a beam splitter network (BSN) to distribute the quantum states across the four nodes. Phase encoding at each node is achieved with a combination of quarter-wave plates (QWPs) and half-wave plates (HWPs). Projective measurements in the σz basis are performed at each node using an HWP and a polarizing beam splitter (PBS), with two-photon coincidence counts measured using superconducting nanowire single-photon detectors (SNSPDs). The theoretical sensitivity bound is calculated using the obtained probabilities, and the experimental sensitivity is determined by calculating the standard deviation of the estimated average phase. The experimental results are compared with the standard quantum limit (SQL) and the Heisenberg scaling (HS). The researchers also present a generalization of their scheme for N photons and d unknown phases, showing that Heisenberg scaling can be achieved even when N<d.

The key findings are: 1. Experimental demonstration of distributed quantum phase sensing among four nodes, achieving a 2.2 dB sensitivity enhancement over the SQL using only a two-photon entangled state. This demonstrates quantum-enhanced sensitivity even with fewer photons than the number of unknown parameters. 2. Theoretical and experimental validation of Heisenberg scaling (HS) in distributed quantum sensing, showing that the sensitivity improves quadratically with the number of photons, even when fewer photons are used than the number of phases to estimate. 3. Development of a scalable scheme for estimating multiple spatially distributed phases (d) using N photons where N can be less than d. The theoretical analysis shows that the Heisenberg scaling can be achieved in this generalized scenario. 4. High visibilities (close to 1) are obtained in the experimental interference fringes. This highlights the high quality of the entangled state generated and the efficiency of the phase encoding and measurement process. 5. The experimental standard deviation of the estimated average phase is close to the theoretically predicted Heisenberg scaling, despite some limitations in experimental conditions (e.g., fiber length, limited number of samples). These limitations are discussed in detail and considered in error analysis.

The results address the long-standing challenge of scalability in distributed quantum sensing by demonstrating quantum enhancement with a reduced number of photons. The achieved sensitivity improvement of 2.2dB over the SQL using only two photons to estimate four phases is significant. This shows the potential to drastically reduce the resource requirements for large-scale distributed quantum sensing networks. The theoretical generalization of the protocol to N photons and d phases further strengthens the claim of scalability. The work significantly advances the field by showing that Heisenberg scaling, the optimal quantum limit, is attainable even when the number of photons is less than the number of parameters, opening up possibilities for practical large-scale implementations of distributed quantum sensing.

This research successfully demonstrates a distributed quantum phase sensing protocol that achieves quantum-enhanced sensitivity using significantly fewer photons than the number of parameters being estimated. The experimental results confirm a 2.2 dB improvement over the standard quantum limit with only two photons estimating four phases. The theoretical analysis extends this finding to a generalized scenario, proving the feasibility of Heisenberg scaling even when the number of photons is less than the number of unknown phases. Future research could focus on improving the experimental setup (reducing noise and losses) to reach the ideal Heisenberg scaling limit and implementing the protocol in more complex and larger distributed sensor networks. Moreover, exploring various linear global functions for estimating multiple phases offers promising future directions.

The current experimental implementation utilizes a post-selection technique, and some experimental imperfections such as photon losses and detector inefficiency were present. While the post-selection does not affect the demonstration of quantum enhancement, future work should consider mitigation of losses to further improve the sensitivity and achieve the full potential of the Heisenberg scaling. The limited number of samples used in the estimation (μ ≈ 367) also contributes to the deviation from the ideal Heisenberg scaling. The fluctuation in phases due to the 3km long optical fibers may also impact the precision of the estimation. Future studies may investigate techniques to improve the stability of the phases over long distances.