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arXiv:2603.20139 · Physical Review Letters

Heisenberg-scaling characterization of a two-channel optical network via two-port homodyne detection

Atmadev Rai, Paolo Facchi, Vincenzo Tamma

quant-phSubmitted Mar 20, 2026• Announced Mar 23, 2026• 0 citations• Revised Jul 5, 2026 (v2)
v1 · Mar 20, 2026v2 · Jul 5, 2026
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Abstract

We present a fully Gaussian and experimentally feasible scheme for the simultaneous estimation of the four real parameters that characterize a two-channel optical network. The scheme utilizes a two-mode squeezed probe and balanced homodyne detection at both output ports, for which we derive the complete classical Fisher information matrix analytically. Our scheme achieves the Heisenberg-scaling sensitivity for all four parameters simultaneously, enabling full multiparameter characterization of the two-channel interferometric network. We further show, by maximum-likelihood estimation, that the corresponding multiparameter Cramér-Rao bounds are saturated with a modest number of experimental repetitions and for low photon number. The scheme establishes a practical route to Heisenberg-scaling multiparameter Gaussian metrology for a two-channel network, with direct relevance to calibration and sensing in integrated photonics and distributed quantum-enhanced measurement architectures.

Journal reference
Phys. Rev. Lett. 137, 010801 (2026)
DOI
10.1103/y9wq-hhg2
Comments
8 Figures

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We present a framework for simultaneously estimating all four real parameters of a general two-channel unitary U(2)\mathrm{U}(2)U(2) with Heisenberg-scaling precision. We derive analytical expressions for the quantum Fisher information matrix and show that all parameters attain the 1/N1/N1/N scaling in the precision by using experimentally feasible Gaussian probes, such as two-mode squeezed states or two single-mode squeezed states. Our results extend multiparameter metrology to its most general two-mode setting and establish concrete design principles for experimental implementations of Heisenberg-scaling, multiparameter optical interferometry with experimentally feasible resources. It not only sheds light on the fundamental interface between quantum interference of squeezed light and quantum metrological advantage in multiparameter estimation, but it also provides an important stepping stone towards the development of a wide range of quantum technologies based on distributed quantum metrology in arbitrary optical networks.
The simultaneous estimation of multiple parameters is a central task in quantum metrology, distributed sensing, and the calibration of large photonic interferometers. A fundamental question is how many independent parameter combinations can inherit Heisenberg scaling from a given number of squeezed probes in a multimode Gaussian network. Here, we answer this question for arbitrary passive linear optical networks. For a ppp-parameter, MMM-channel interferometer probed by kkk single-mode squeezed states and at least one coherent state in the remaining channels, we show that the rank of the Heisenberg-scaling coefficient of the quantum Fisher information matrix is bounded by nHS≤min⁡{p,k(k+3)/2}n_{\rm HS}\le \min\{p,k(k+3)/2\}nHS​≤min{p,k(k+3)/2}, which corresponds to the maximum number of independent combinations of parameters that can be estimated with Heisenberg-scaling sensitivity. The bound separates into two geometrically distinct contributions. The covariance contribution of the quantum Fisher information, which describes squeezing-enhanced fluctuations, provides at most k(k+1)/2k(k+1)/2k(k+1)/2 parameter combinations estimable at Heisenberg-scaling sensitivity, while the first-moment contribution provides at most kkk additional independent parameter combinations with Heisenberg-scaling sensitivity. We identify the conditions for saturating these bounds and construct a passive family of interferometers that saturates these bounds.
Quantum metrology enables parameter estimation beyond classical limits by exploiting nonclassical resources such as squeezing and entanglement. In distributed quantum sensing, Heisenberg scaling has been extended from 1/N21/N^21/N2 to 1/(NM)21/(NM)^21/(NM)2 through entanglement across both particles and spatial modes, where NNN denotes the photon number and MMM the number of spatially distributed modes. However, the overall sensitivity has remained limited to linear scaling with the number of measurement repetitions RRR. Here, we show that exploiting entanglement across temporal modes via time-domain multiplexing enables a scaling advantage with respect to RRR. As a result, the sensitivity can asymptotically approach simultaneous Heisenberg scaling in photons, spatial modes, and repetitions, yielding an overall sensitivity approaching Δ2φ∝1/(NMR)2Δ^2 φ\propto 1/(NMR)^2Δ2φ∝1/(NMR)2. Using the Bogoliubov transformation formalism, we prove the optimality of the protocol within the class of Gaussian states and show that the scaling is realizable via homodyne detection and maximum-likelihood estimation. We further show that the advantage persists under optical loss and propose an experimentally feasible loop-based photonic sensing scheme. Our results open a route to incorporating time-multiplexing techniques into quantum metrology.
Abstract This work presents a quantum sensing interferometric scheme for the simultaneous estimation of multiple parameters at the ultimate quantum scaling precision in a two-mode optical network. This scheme is experimentally feasible because it employs input Gaussian states such as squeezed and coherent states and a relatively low number of experimental runs. We focus on the precise and simultaneous measurement of two unknown phase shifts and an unknown beam splitter reflectivity, achieving Heisenberg scaling sensitivity in all three parameters without any constraints on the parameter values.
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
Private distributed quantum sensing aims to estimate an authorized collective parameter while preventing independent estimation of individual local parameters. Here, we analytically characterize Gaussian quantum networks satisfying this perfect local privacy condition. Using a graph representation of the Gaussian pairing matrix, we show that two-mode squeezed vacuum states constitute the essential building blocks of privacy-preserving Gaussian states. We then analytically derive the optimal sensitivity under perfect local privacy and determine a Gaussian probe that achieves Heisenberg scaling with respect to both the photon number and the number of sensing modes. Using local homodyne measurements and maximum-likelihood estimation, we numerically verify quantum-enhanced sensitivity beyond the shot-noise limit while preserving perfect local privacy. We further show that the local-privacy condition is preserved under arbitrary phase-independent quantum channels, including optical loss. Our results provide a general framework for constructing and optimizing privacy-preserving continuous-variable quantum sensing networks.