Heisenberg scaling precision in multi-mode distributed quantum metrology

Abstract

We consider the estimation of an arbitrary parameter φ, such as the temperature or a magnetic field, affecting in a distributed manner the components of an arbitrary linear optical passive network, such as an integrated chip. We demonstrate that Heisenberg scaling precision (i.e. of the order of 1/N, where N is the number of probe photons) can be achieved without any iterative adaptation of the interferometer hardware and by using only a simple, single, squeezed light source and well-established homodyne measurements techniques. Furthermore, no constraint on the possible values of the parameter is needed but only a preliminary shot-noise estimation (i.e. with a precision of N ) easily achievable without any quantum resources. Indeed, such a classical knowledge of the parameter is enough to prepare a single, suitable optical stage either at the input or the output of the network to monitor with Heisenberg-limited precision any variation of the parameter to the order of 1 / N without the need to iteratively modify such a stage.

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.…”

“…The role of the two auxiliary stages and is to respectively distribute the photons of the probe through multiple channels, and then to refocus them into the only observed channel. We will show that only one auxiliary network needs to be optimized to reach the Heisenberg scaling, while, for networks with a large number of channels, the effect of the non-optimized network is typically irrelevant on the overall precision of the estimation [ 42 ]. Reprinted with permission from ref.…”

“…This review is organised in two parts, in which we introduce two distinct estimation schemes, discussing the features that differentiate the two approaches. In Section 2 , we will discuss a squeezing-encoding scheme achieving Heisenberg-scaling sensitivity by employing a single squeezed vacuum state and homodyne detection at a single output channel [ 42 ]. We will assume that no prior knowledge on the value of the parameter is known, and that the structure of the network, as well as the nature of the parameter it encodes, are completely arbitrary.…”

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.…”