Non-adaptive Heisenberg-limited metrology with multi-channel homodyne measurements

Abstract

We show a protocol achieving the ultimate Heisenberg-scaling sensitivity in the estimation of a parameter encoded in a generic linear network, without employing any auxiliary networks, and without the need of any prior information on the parameter nor on the network structure. As a result, this protocol does not require a prior coarse estimation of the parameter, nor an adaptation of the network. The scheme we analyse consists of a single-mode squeezed state and homodyne detectors in each of the M output channels of the network encoding the parameter, making it feasible for experimental applications.

Estimation with Heisenberg-Scaling Sensitivity of a Single Parameter Distributed in an Arbitrary Linear Optical Network

“…In this work, we have reviewed in detail two schemes which address these limitations [ 42 , 43 ]. By employing analyses based on the Cramér–Rao bound, i.e., the ultimate precision achievable for a given estimation scheme, and on the Fisher information, we were able to assess the super-sensitivity of various feasible metrological setups, always achievable in the regime of large statistical samples through the maximum-likelihood estimator.…”

“… Example of a passive and linear network which depends on a single global parameter . The parameter can be thought of as a physical property of an external agent (e.g., temperature, electromagnetic field) which affects multiple components, possibly of different natures, of the network [ 42 , 43 ]. Reprinted with permission from ref.…”

“…These results show that it is possible to conceive feasible two-step estimation strategies, composed of a first classical estimation of the unknown parameter required to engineer the auxiliary stage, and then the actual quantum estimation. In Section 3 , we describe a different scheme achieving the Heisenberg scaling, which makes use of a single-mode squeezed-coherent state as a probe, and homodyne detection in every output port of the linear network [ 43 ]. Once again, the model will assume no prior knowledge on the parameter, nor on the structure of the network.…”

The Role of Auxiliary Stages in Gaussian Quantum Metrology

“…In recent years, much attention has been put in the study of metrological schemes that exploit quantum resources, such as entanglement and squeezing, to enhance the sensitivity in the estimation of physical properties beyond the possibilities of classical strategies, with applications to imaging [1,2], thermometry [3,4], mapping of magnetic fields [5,6] and gravitational waves detection [7], among others. One of the most emblematic quantum enhancements sought in quantum metrology is the renown Heisenberg limit , which consists in achieving a scaling of the estimation error in the number N of probes (typically photons, or atoms) of order of 1/N, which surpasses the classical (or shot-noise) limit 1/ √ N. Gaussian metrology, which specializes in the study of estimation schemes employing Gaussian states of light and squeezing as metrological resource [29][30][31][32], represents a promising path towards a feasible quantum-enhancement in estimation strategies and the Heisenberg-scaling sensitivity [33][34][35][36][37][38][39][40][41]. It exploits the possibility to reduce the intrinsic noise of the electromagnetic field quadratures below the quantum fluctuations of the vacuum.…”

“…Such a reduced noise, together with relatively easy-to-implement experimental procedures to produce these squeezed-noise states, and their increased robustness to decoherence compared to entangled states, make the Gaussian approach of great interest for short-term applications of quantum technologies. A particular case analysed by quantum metrology is the estimation of a single unknown parameter that appears within a given optical linear network multiple times, affecting for example different interferometric components [34][35][36]38,[42][43][44][45][46][47][48] (see Figure 1). This is the case of unknown temperatures or magnitudes of the electromagnetic field, which modifies the physical properties of the optical parts composing the network within the regime of passive and linear evolution of the probe.…”