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Bulletin of the American Physical Society

Scalable Heisenberg limited metrology using mixed states

Geng Chen, N. Aharon, Yong-Nan Sun, Zi-Huai Zhang, Wen-Hao Zhang, De‐Yong He, Jian‐Shun Tang, Yaron Kedem, Chuan‐Feng Li, Guang‐Can Guo

Published Dec 22, 2016• 0 citations
OpenAlex

Abstract

Improving the precision of measurements is a prime challenge to the scientific community. Quantum metrology provides methods to overcome the standard quantum limit (SQL) of 1/sqrt{N} and to reach the fundamental Heisenberg-limit (HL) of 1/N. While a lot of theoretical and experimenta works have been dedicated to this task, most of the attempts focused on utilizing NOON and squeezed states, which exhibit unique quantum correlations. Here we present, and experimentally implement, a new scheme for precision measurements that enables reaching the HL. Our scheme is based on a probe in a mixed state with a large uncertainty, combined with a post-selection, such that the Fisher information is maximized, and the Carmer-Rao bound is saturated. We performed a Heisenberg limited measurement of the Kerr non-linearity of a single photon, where an ultra-small Kerr phase of 6 *10^{-8}$ was observed with an unprecedented precision of 10^{-9}. Our method paves the way to Heisenberg-limited metrology with only mixed state.

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Quantum metrology allows the enhanced precision on estimating phase shifts in an optical interferometer to reach the $\ensuremath{\propto}1/N$ scaling of precision ($N$ is the mean photon number of the light field). But when $N$ is very large, the nonlinear optical effects may be nonnegligible. Here we investigate theoretically the quantum uncertainty bounds for simultaneously detecting the linear and nonlinear phase shifts. Focusing our attentions only on the linear and the nonlinear Kerr phase shifts, we find the NOON states can not be used to measure the two phase parameters. Using the entangled coherent states, the estimation precision is limited by the $1/{N}^{1/2}$ ($1/{N}^{3/2}$) scaling for the linear (nonlinear) phase. To increase the precisions and beat the standard quantum limit, we show that the entangled Schr\"odinger-cat states can be used to reach the $1/N$ scaling for the linear phase and $1/{N}^{2}$ scaling for the nonlinear phase.

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Improving the precision of measurements is a prime challenge to the scientific community. Quantum metrology provides methods to overcome the standard quantum limit (SQL) of 1/sqrt{N} and to reach the fundamental Heisenberg-limit (HL) of 1/N. While a lot of theoretical and experimenta works have been dedicated to this task, most of the attempts focused on utilizing NOON and squeezed states, which exhibit unique quantum correlations. Here we present, and experimentally implement, a new scheme for precision measurements that enables reaching the HL. Our scheme is based on a probe in a mixed state with a large uncertainty, combined with a post-selection, such that the Fisher information is maximized, and the Carmer-Rao bound is saturated. We performed a Heisenberg limited measurement of the Kerr non-linearity of a single photon, where an ultra-small Kerr phase of 6 *10^{-8}$ was observed with an unprecedented precision of 10^{-9}. Our method paves the way to Heisenberg-limited metrology with only mixed state.
Abstract Improving the precision of measurements is a significant scientific challenge. Previous works suggest that in a photon-coupling scenario the quantum fisher information shows a quantum-enhanced scaling of N 2 , which in theory allows a better-than-classical scaling in practical measurements. In this work, utilizing mixed states with a large uncertainty and a post-selection of an additional pure system, we present a scheme to extract this amount of quantum fisher information and experimentally attain a practical Heisenberg scaling. We performed a measurement of a single-photon’s Kerr non-linearity with a Heisenberg scaling, where an ultra-small Kerr phase of ≃6 × 10 −8 rad was observed with a precision of ≃3.6 × 10 −10 rad. From the use of mixed states, the upper bound of quantum fisher information is improved to 2 N 2 . Moreover, by using an imaginary weak-value the scheme is robust to noise originating from the self-phase modulation.
Abstract Improving the precision of measurements is a significant scientific challenge. Previous works suggest that in a photon-coupling scenario the quantum fisher information shows a quantum-enhanced scaling of N2, which in theory allows a better-than-classical scaling in practical measurements. In this work, utilizing mixed states with a large uncertainty and a post-selection of an additional pure system, we present a scheme to extract this amount of quantum fisher information and experimentally attain a practical Heisenberg scaling. We performed a measurement of a single-photon’s Kerr non-linearity with a Heisenberg scaling, where an ultra-small Kerr phase of ≃6 × 10−8 rad was observed with a precision of ≃3.6 × 10−10 rad. From the use of mixed states, the upper bound of quantum fisher information is improved to 2N2. Moreover, by using an imaginary weak-value the scheme is robust to noise originating from the self-phase modulation.
The use of quantum resources can provide measurement precision beyond the shot noise limit (SNL). The task of ab initio optical phase measurement-the estimation of a completely unknown phase-has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able-even in principle-to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techniques: entanglement, multiple samplings of the phase shift, and adaptive measurement. Our experimental demonstration of the scheme, with photonic qubits and N = 3 photon-passes, achieves a precision that is within 4% of the HL, surpassing the best precision theoretically achievable with simpler techniques. This work represents a fundamental achievement of the ultimate limits of metrology, and the scheme can be extended to higher N and other physical systems.
Optical phase measurement through its application in quantum metrology has pushed the precision limit with which some physical quantities can be measured accurately. At the very fundamental level, the laws of quantum mechanics dictate that the uncertainty in phase estimations scales as 1/N, where N is the number of quantum resources employed in the protocol [1]. This is the well known Heisenberg limit (HL) which is quadratically better than the traditional precision limit known as the standard quantum limit (SQL) with uncertainty asymptotically scaling as 1/sqrt{N} [1]. Several experiments have demonstrated that the SQL can be beaten by using an entangled state as the probe and a specific measurement scheme for ab initio estimation of unknown phases [2,3]. It has also been shown experimentally that even in the absence of the entanglement one can measure an unknown phase with imprecision scaling at the HL [4]. In this work we first present a new protocol able to estimate an optical phase at the Heisenberg limit, and then experimentally explore fundamental and practical issues in generating high-quality novel entangled states, for use in this protocol and beyond. Our aim in this study is to measure an unknown phase in the interval [0,2pi) with uncertainty attaining the exact HL. There is a condition that should be met to address this objective: preparation of an optimal state [5]. This would cover part of the presentation through which we explain how to experimentally realise such an optimal state with the current technological limitations and the feasibility of the scheme. In particular, we generate an entangled 3-photon (2-photon) state of specific superposition of GHZ (Bell) states. Our numerical simulation of the phase measurement gate together with the experimental outcomes show that the created state should have a high fidelity and purity to be able to have the phase uncertainty achieving the exact HL. Therefore, we briefly explain the modelling for experimental imperfections and finally present the results of experimental phase measurements.
It is well known that classical states of light exhibit shot noise, characteristic of independent or uncorrelated particles. For phase estimation problems, this leads to a shot-noise limited uncertainty of 1/sqrt[N], where N is the number of particles detected. It is also well known that the shot-noise limit is not fundamental: squeezed states and entangled states can be used for sub-shot-noise phase measurements. The uncertainty principle sets a fundamental limit of 1/N, known as the "Heisenberg" limit. We have recently demonstrated a method, using parametric downconversion and post-selection, to generate entangled "NOON" states suitable for sub-shot-noise phase measurements [M.W. Mitchell et al, Nature 429, 161 (2004)]. We generated a three-photon NOON state and demonstrated three-fold improvement in phase resolution with this state. The relationship between phase resolution and phase uncertainty depends on prior information about the phase being estimated. As in the case of phase measurements with squeezed states, extra precision in one dimension is gained at the cost of reduced precision in other dimensions. Only when prior information is incorporated can entangled-state metrology be applied to beat the shot-noise limit. We illustrate this relationship and discuss adaptive strategies for phase estimation and the possibility of reaching the Heisenberg limit.
Quantum metrology allows the enhanced precision on estimating phase shifts in an optical interferometer to reach the $\ensuremath{\propto}1/N$ scaling of precision (NNN is the mean photon number of the light field). But when NNN is very large, the nonlinear optical effects may be nonnegligible. Here we investigate theoretically the quantum uncertainty bounds for simultaneously detecting the linear and nonlinear phase shifts. Focusing our attentions only on the linear and the nonlinear Kerr phase shifts, we find the NOON states can not be used to measure the two phase parameters. Using the entangled coherent states, the estimation precision is limited by the 1/N1/21/{N}^{1/2}1/N1/2 (1/N3/21/{N}^{3/2}1/N3/2) scaling for the linear (nonlinear) phase. To increase the precisions and beat the standard quantum limit, we show that the entangled Schr\"odinger-cat states can be used to reach the 1/N1/N1/N scaling for the linear phase and 1/N21/{N}^{2}1/N2 scaling for the nonlinear phase.