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arXiv:2607.24657

Geometric bounds on multiparameter Heisenberg scaling in optical metrology with limited squeezed resources

Atmadev Rai, Paolo Facchi, Vincenzo Tamma

quant-phSubmitted Jul 27, 2026• Announced Jul 28, 2026• 0 citations
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Abstract

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 $p$-parameter, $M$-channel interferometer probed by $k$ 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 $n_{\rm HS}\le \min\{p,k(k+3)/2\}$, 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 $k(k+1)/2$ parameter combinations estimable at Heisenberg-scaling sensitivity, while the first-moment contribution provides at most $k$ 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.

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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.
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.
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.
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.
We identify precision limits for the simultaneous estimation of multiple parameters in multimode interferometers. Quantum strategies to enhance the multiparameter sensitivity are based on entanglement among particles, modes, or combining both. The maximum attainable sensitivity of particle-separable states defines the multiparameter shot-noise limit, which can be surpassed without mode entanglement. Further enhancements up to the multiparameter Heisenberg limit are possible by adding mode entanglement. Optimal strategies that saturate the precision bounds are provided.
Quantum sensing and quantum metrology propose schemes for the estimation of physical properties, such as lengths, time intervals, and temperatures, achieving enhanced levels of precision beyond the possibilities of classical strategies. However, such an enhanced sensitivity usually comes at a price: the use of probes in highly fragile states, the need to adaptively optimise the estimation schemes to the value of the unknown property we want to estimate, and the limited working range, are some examples of challenges which prevent quantum sensing protocols to be practical for applications. This work reviews two feasible estimation schemes which address these challenges, employing easily realisable resources, i.e., squeezed light, and achieve the desired quantum enhancement of the precision, namely the Heisenberg-scaling sensitivity. In more detail, it is here shown how to overcome, in the estimation of any parameter affecting in a distributed manner multiple components of an arbitrary M-channel linear optical network, the need to iteratively optimise the network. In particular, we show that this is possible with a single-step adaptation of the network based only on a prior knowledge of the parameter achievable through a “classical” shot-noise limited estimation strategy. Furthermore, homodyne measurements with only one detector allow us to achieve Heisenberg-limited estimation of the parameter. We further demonstrate that one can avoid the use of any auxiliary network at the price of simultaneously employing multiple detectors.
Quantum mode parameter estimation determines parameters governing the shape of electromagnetic modes occupied by a quantum state of radiation. Canonical examples, time delays and frequency shifts, underpin radar, lidar, and optical clocks. A comprehensive framework recently established that broad families of quantum states can attain the Heisenberg limit, surpassing any classical strategy. This raises a fundamental question: among all quantum-enhanced strategies, which is truly optimal? Answering this requires identifying physically meaningful resources governing each estimation task, so quantum states can be compared on equal footing. We show these resources are connected to the eigenmode basis of the generator of the relevant mode transformation. For time-shift estimation, whose generator is diagonal in the frequency domain, the pertinent resources are the mean frequency and bandwidth; analogous quantities emerge for other transformations. Our framework unifies two historically separate perspectives: the particle-number aspect and the mode-structure of quantum light, providing a coherent picture of quantum-enhanced sensing with multimode radiation. Within this unified framework, we derive a tight upper bound on the quantum Fisher information for multimode Gaussian states, expressed in terms of these natural resources, and analytically identify the optimal Gaussian states saturating it. These optimal states take a particularly transparent form in the generator eigenbasis, a structural simplicity reflecting the deep connection between the geometry of the mode transformation and the architecture of the optimal probe. We further demonstrate that multimode homodyne detection constitutes the optimal measurement, achieving this bound and completing the end-to-end characterization of optimal quantum metrology strategies for mode parameter estimation.