Parameter estimation at the fundamental limit given by the quantum Cramér-Rao bound is an important milestone for quantum metrology. Here, we present experiments that saturate this bound for transmission estimation with a macroscopic quantum state.
The field of quantum metrology seeks to apply quantum techniques and/or resources to classical sensing approaches with the goal of enhancing the precision in the estimation of a parameter beyond what can be achieved with classical resources. Theoretically, the fundamental minimum uncertainty in the estimation of a parameter for a given probing state is bounded by the quantum Cramér-Rao bound. From a practical perspective, it is necessary to find physical measurements that can saturate this fundamental limit and to show experimentally that it is possible to perform measurements with the required precision to do so. Here we perform experiments that saturate the quantum Cramér-Rao bound for transmission estimation over a wide range of transmissions when probing the system under study with a continuous wave bright two-mode squeezed state. To properly take into account the imperfections in the generation of the quantum state, we extend our previous theoretical results to incorporate the measured properties of the generated quantum state. For our largest transmission level of 84%, we show a 62% reduction over the optimal classical protocol in the variance in transmission estimation when probing with a bright two-mode squeezed state with -8 dB of intensity-difference squeezing. Given that transmission estimation is an integral part of many sensing protocols, such as plasmonic sensing, spectroscopy, calibration of the quantum efficiency of detectors, etc., the results presented promise to have a significant impact on a number of applications in various fields of research.
The minimum possible uncertainty when estimating a parameter, in this case transmission, is bounded by the quantum Cramér-Rao bound (QCRB). This bound is independent of the measurement technique and as result can be used to quantify how close a measurement is to the optimal one for a given system and probing state, as an optimal measurement saturates the QCRB. The ability to perform transmission measurements through an optical system at the QCRB can lead to enhancements in the calibration of optimal states for interferometers, the characterization of high efficiency photodetectors, or the sensitivity of sensing devices based on transmission, such as plasmonic sensors or spectroscopy. We show theoretically that the ultimate sensitivity limit per photon in the estimation of transmission, achievable with Fock states or vacuum two-mode squeezed states (vTMSS), can be approached through the use of bright quantum states of light, such as bright single-mode squeezed states (bSMSS) or bright two-mode squeezed states (bTMSS). The use of bright quantum states has the advantage that they can be generated with a significantly larger number of photons, which leads to an absolute sensitivity that can be many orders of magnitude larger than the one achieved with Fock states or vTMSS. We identify a measurement strategy that saturates the QCRB for these bright quantum states and shown experimentally that transmission estimation at the QCRB can be achieved with bTMSS through optimized intensity difference measurements.
The power of quantum sensing rests on its ultimate precision limit, quantified by the quantum Cramer-Rao bound (QCRB), which can surpass classical bounds. In multi-parameter estimation, the QCRB is not always saturated as the quantum nature of associated observables may lead to their incompatibility. Here we explore the precision limits of multi-parameter estimation through the lens of quantum geometry, enabling us to experimentally evaluate the QCRB via quantum geometry measurements. Focusing on two- and three-parameter estimation, we elucidate how fundamental quantum uncertainty principles prevent the saturation of the bound. By linking a metric of "quantumness" to the system geometric properties, we investigate and experimentally extract the attainable QCRB for three-parameter estimations.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
Abstract Sensing and imaging are among the most important applications of quantum information science. To investigate their fundamental limits and the possibility of quantum enhancements, for decades researchers have relied on the quantum Cramér-Rao lower error bounds pioneered by Helstrom. Recent work, however, has called into question the tightness of those bounds for highly nonclassical states in the non-asymptotic regime, and better methods are now needed to assess the attainable quantum limits in reality. Here we propose a new class of quantum bounds called quantum Weiss-Weinstein bounds, which include Cramér-Rao-type inequalities as special cases but can also be significantly tighter to the attainable error. We demonstrate the superiority of our bounds through the derivation of a Heisenberg limit and phase estimation examples.