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

In multiparameter quantum metrology, researchers often aim to estimate multiple unknown phases simultaneously using photonic networks. A critical challenge is that nonclassical resources, such as squeezed states, are limited. This paper investigates the fundamental limit on how many independent parameter combinations can achieve Heisenberg-scaling precision—where sensitivity scales as O(1/N^2)—given a fixed number of squeezed probes in a passive multimode Gaussian network.

The authors analyze the Quantum Fisher Information Matrix (QFIM) for a p-parameter, M-channel passive linear optical network. They decompose the QFIM into two distinct geometric contributions: the covariance contribution (representing squeezing-enhanced fluctuations) and the first-moment contribution (representing displacements of the Gaussian center). By examining the rank of these contributions within the active squeezed input subspace, the authors derive a parameterization-independent bound on the dimension of the Heisenberg-scaling subspace.

The study establishes that the total number of independent parameter combinations capable of Heisenberg scaling is bounded by n_HS ≤ min{p, k(k+3)/2}. The covariance contribution provides at most k(k+1)/2 directions, while the first-moment contribution adds at most k directions. The authors prove these bounds are tight by constructing a family of passive interferometers that saturate the limit. Furthermore, they determine that to estimate all p parameters with Heisenberg-scaling precision, one requires at least k_min ≈ sqrt(8p) squeezed states.

This work provides a clear resource-counting principle for large-scale photonic sensors. It demonstrates that simply increasing the number of modes or tunable elements in a network does not inherently increase the number of parameters that can be measured with Heisenberg-scaling precision. Instead, the performance is fundamentally constrained by the geometry of how squeezed resources couple to the encoded parameters. These bounds serve as a benchmark for designing efficient quantum sensing architectures.