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arXiv:2208.00011 · New Journal of Physics

Multiparameter transmission estimation at the quantum Cramér–Rao limit on a cloud quantum computer

Aaron Z. Goldberg, Khabat Heshami

Published Nov 1, 2022• 6 citations
arXivHTMLPDF

Abstract

Abstract Estimating transmission or loss is at the heart of spectroscopy. To achieve the ultimate quantum resolution limit, one must use probe states with definite photon number and detectors capable of distinguishing the number of photons impinging thereon. In practice, one can outperform classical limits using two-mode squeezed light, which can be used to herald definite-photon-number probes, but the heralding is not guaranteed to produce the desired probes when there is loss in the heralding arm or its detector is imperfect. We show that this paradigm can be used to simultaneously measure distinct loss parameters in both modes of the squeezed light, with attainable quantum advantages. We demonstrate this protocol on Xanadu’s X8 chip, accessed via the cloud, building photon-number probability distributions from 106 shots and performing maximum likelihood estimation (MLE) on these distributions 103 independent times. Because pump light may be lost before the squeezing occurs, we also simultaneously estimate the actual input power, using the theory of nuisance parameters. MLE converges to estimate the transmission amplitudes in X8’s eight modes to be 0.392 02(6), 0.307 06(8), 0.369 37(6), 0.287 30(9), 0.382 06(6), 0.304 41(8), 0.372 29(6), and 0.286 21(8) and the squeezing parameters, which are proxies for effective input coherent-state amplitudes, their losses, and their nonlinear interaction times, to be 1.3000(2), 1.3238(3), 1.2666(2), and 1.3425(3); all of these uncertainties are within a factor of two of the quantum Cramér–Rao bound. This study provides crucial insight into the intersection of quantum multiparameter estimation theory, MLE convergence, and the characterization and performance of real quantum devices.

DOI
10.1088/1367-2630/aca21c

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Abstract Estimating transmission or loss is at the heart of spectroscopy. To achieve the ultimate quantum resolution limit, one must use probe states with definite photon number and detectors capable of distinguishing the number of photons impinging thereon. In practice, one can outperform classical limits using two-mode squeezed light, which can be used to herald definite-photon-number probes, but the heralding is not guaranteed to produce the desired probes when there is loss in the heralding arm or its detector is imperfect. We show that this paradigm can be used to simultaneously measure distinct loss parameters in both modes of the squeezed light, with attainable quantum advantages. We demonstrate this protocol on Xanadu’s X8 chip, accessed via the cloud, building photon-number probability distributions from 106 shots and performing maximum likelihood estimation (MLE) on these distributions 103 independent times. Because pump light may be lost before the squeezing occurs, we also simultaneously estimate the actual input power, using the theory of nuisance parameters. MLE converges to estimate the transmission amplitudes in X8’s eight modes to be 0.392 02(6), 0.307 06(8), 0.369 37(6), 0.287 30(9), 0.382 06(6), 0.304 41(8), 0.372 29(6), and 0.286 21(8) and the squeezing parameters, which are proxies for effective input coherent-state amplitudes, their losses, and their nonlinear interaction times, to be 1.3000(2), 1.3238(3), 1.2666(2), and 1.3425(3); all of these uncertainties are within a factor of two of the quantum Cramér–Rao bound. This study provides crucial insight into the intersection of quantum multiparameter estimation theory, MLE convergence, and the characterization and performance of real quantum devices.
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.
Enhancing the precision of a measurement requires maximizing the information that can be gained about the quantity of interest from probing a system. For optical-based measurements, such an enhancement can be achieved through two approaches, increasing the number of photons used to interrogate the system and using quantum states of light to increase the amount of quantum Fisher information gained per photon. Here we consider the use of quantum states of light with a large number of photons, namely the bright single-mode squeezed state and the bright two-mode squeezed state, which take advantage of both of these approaches for the problem of transmission estimation. We show that, in the limit of large squeezing, these states approach the maximum possible quantum Fisher information per photon for transmission estimation that is achieved with the Fock state and the vacuum two-mode squeezed state. Since the bright states we consider can be generated at powers much higher than those of the quantum states that achieve the maximum quantum Fisher information per photon, they can achieve a much higher absolute precision as quantified by the quantum Cram\'er-Rao bound. We discuss the effects of losses external to the system on the precision of transmission estimation and identify simple measurement techniques that can saturate the quantum Cram\'er-Rao bound for the bright squeezed states even in the presence of such external losses.
The multiparameter estimation in quantum metrology is significant for a large number of applications for quantum information tasks. Nevertheless, how mode correlations affect the multiparameter-estimation precision with and without photon-loss cases remains ambiguous. For this reason, here, we present the relationship between the quantum Cram\'er-Rao bound and the mode correlations in the framework of the simultaneous two-phase estimation for a three-mode Mach-Zehnder interferometer induced by the tritter which is composed of three cascaded beam splitters and phase shifters. By choosing different input resources, including a two-mode squeezed vacuum state $\ensuremath{\bigotimes}$ a coherent state (scheme I) and a twin squeezed vacuum state $\ensuremath{\bigotimes}$ a coherent state (scheme II), the results show that compared with scheme II, scheme I shows a better estimation performance even in the presence of photon losses. The reason may be that the intramode correlation of the probe state for scheme I is stronger than that for scheme II, while the intermode correlation of the probe state for the former is weaker than that for the latter. This allow us to open vital perspectives on the essence of enhanced multiparameter-estimation precision.
Recently, there has been significant interest in multiple-parameter quantum estimation techniques that exploit quantum resources. In particular, the estimation of optical transmission is a crucial parameter in various scientific fields and industries. In optical-based sensing, precision can be enhanced through two approaches: increasing the number of photons that interact with the samples and utilizing quantum states of light. In this paper, we investigate multiple transmission estimation using bright two-mode squeezed states (bTMSSs), which combine the advantages of both bright light and quantum states. We calculate the precision bound for multiple transmission estimation by employing multiple copies of bTMSSs with experimentally feasible optimal measurement, determining both the quantum Cramer-Rao bound and the Cramer-Rao bound. Our results demonstrate that multiple copies of bTMSSs can achieve quantum-enhanced sensitivity for multiple transmissions compared to coherent states, and the ultimate limit of precision can be attained in regions with high nonlinear gain. Furthermore, as an application, we show the quantum-enhanced sensing in circular dichroism sensing using a pair of bTMSSs. Our strategy for multiple transmission estimation offers a practical platform for exploring real-world quantum sensing applications.
Abstract We discuss the ultimate precision bounds on the multiparameter estimation of single- and two-mode pure Gaussian states. By leveraging on previous approaches that focused on the estimation of a complex displacement only, we derive the Holevo Cramér–Rao bound (HCRB) for both displacement and squeezing parameter characterizing single and two-mode squeezed states. In the single-mode scenario, we obtain an analytical bound and find that it degrades monotonically as the squeezing increases. Furthermore, we prove that heterodyne detection is nearly optimal in the large squeezing limit, but in general the optimal measurement must include non-Gaussian resources. On the other hand, in the two-mode setting, the HCRB improves as the squeezing parameter grows and we show that it can be attained using double-homodyne detection.