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Physical Review Letters

“Super-Heisenberg” and Heisenberg Scalings Achieved Simultaneously in the Estimation of a Rotating Field

Zhibo Hou, Jin Yan, Hongzhen Chen, Jun-Feng Tang, Chang-Jiang Huang, Haidong Yuan, Guo‐Yong Xiang, Chuan‐Feng Li, Guang‐Can Guo

Published Feb 18, 2021• 62 citations
Publisher

Abstract

The Heisenberg scaling, which scales as N^{-1} in terms of the number of particles or T^{-1} in terms of the evolution time, serves as a fundamental limit in quantum metrology. Better scalings, dubbed as "super-Heisenberg scaling," however, can also arise when the generator of the parameter involves many-body interactions or when it is time dependent. All these different scalings can actually be seen as manifestations of the Heisenberg uncertainty relations. While there is only one best scaling in the single-parameter quantum metrology, different scalings can coexist for the estimation of multiple parameters, which can be characterized by multiple Heisenberg uncertainty relations. We demonstrate the coexistence of two different scalings via the simultaneous estimation of the magnitude and frequency of a field where the best precisions, characterized by two Heisenberg uncertainty relations, scale as T^{-1} and T^{-2}, respectively (in terms of the standard deviation). We show that the simultaneous saturation of two Heisenberg uncertainty relations can be achieved by the optimal protocol, which prepares the optimal probe state, implements the optimal control, and performs the optimal measurement. The optimal protocol is experimentally implemented on an optical platform that demonstrates the saturation of the two Heisenberg uncertainty relations simultaneously, with up to five controls. As the first demonstration of simultaneously achieving two different Heisenberg scalings, our study deepens the understanding on the connection between the precision limit and the uncertainty relations, which has wide implications in practical applications of multiparameter quantum estimation.

DOI
10.1103/physrevlett.126.070503

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The Heisenberg scaling, which scales as N^{-1} in terms of the number of particles or T^{-1} in terms of the evolution time, serves as a fundamental limit in quantum metrology. Better scalings, dubbed as "super-Heisenberg scaling," however, can also arise when the generator of the parameter involves many-body interactions or when it is time dependent. All these different scalings can actually be seen as manifestations of the Heisenberg uncertainty relations. While there is only one best scaling in the single-parameter quantum metrology, different scalings can coexist for the estimation of multiple parameters, which can be characterized by multiple Heisenberg uncertainty relations. We demonstrate the coexistence of two different scalings via the simultaneous estimation of the magnitude and frequency of a field where the best precisions, characterized by two Heisenberg uncertainty relations, scale as T^{-1} and T^{-2}, respectively (in terms of the standard deviation). We show that the simultaneous saturation of two Heisenberg uncertainty relations can be achieved by the optimal protocol, which prepares the optimal probe state, implements the optimal control, and performs the optimal measurement. The optimal protocol is experimentally implemented on an optical platform that demonstrates the saturation of the two Heisenberg uncertainty relations simultaneously, with up to five controls. As the first demonstration of simultaneously achieving two different Heisenberg scalings, our study deepens the understanding on the connection between the precision limit and the uncertainty relations, which has wide implications in practical applications of multiparameter quantum estimation.
Quantum estimation of a single parameter has been studied extensively. Practical applications, however, typically involve multiple parameters, for which the ultimate precision is much less understood. Here, by relating the precision limit directly to the Heisenberg uncertainty relation, we show that to achieve the highest precisions for multiple parameters at the same time requires the saturation of multiple Heisenberg uncertainty relations simultaneously. Guided by this insight, we experimentally demonstrate an optimally controlled multipass scheme, which saturates three Heisenberg uncertainty relations simultaneously and achieves the highest precisions for the estimation of all three parameters in SU(2) operators. With eight controls, we achieve a 13.27-dB improvement in terms of the variance (6.63 dB for the SD) over the classical scheme with the same loss. As an experiment demonstrating the simultaneous achievement of the ultimate precisions for multiple parameters, our work marks an important step in multiparameter quantum metrology with wide implications.
Quantum metrology achieves Heisenberg-limited precision in single-parameter estimation, but its multiparameter extension is fundamentally constrained by both parameter-encoding and measurement incompatibility. Noncommuting signal generators may cause incompatible parameter-encoding, preventing the quantum Fisher information matrix from simultaneously achieving the Heisenberg scale for all parameters. Due to incompatible optimal measurements, the classical Fisher information matrix represents the practical attainable precision. Here, we introduce a multiparameter metrology framework based on indefinite evolution (IE), in which different control operations and signal reversal are placed in a coherent superposition. For a single-qubit probe with mutually orthogonal signal generators, IE enables compatible parameter encoding and optimal measurement without the signal reversal. For parallel generators, where only signal reversal realized by its generator is available, IE can achieve the same performance. We further extend this mechanism to noisy, many-body, and high-dimensional probes, and establish general conditions for achieving the simultaneous Heisenberg-limit. In contrast, definite evolution cannot achieve the same performance under compatible optimal measurements, even when signal reversal is available. Our results identify IE as an operational resource for overcoming multiparameter incompatibility and open a route toward attainable Heisenberg-limited sensing in interferometric platforms.
Abstract The Heisenberg limit is acknowledged as the ultimate precision limit for parameter estimation in quantum metrology. Recent studies have demonstrated super-Heisenberg-scaling limits, where the root mean square errors of estimation decrease faster-than-linearly with the number of probes or evolution processes. This casts doubt on what the ultimate precision limit of quantum metrology really is and, in particular, whether there is a deeper connection with Heisenberg's uncertainty principle. By linking the dynamics of the metrological process with properties in parameter space, we construct a Heisenberg's uncertainty relation between the unknown parameter and its canonical momentum. We propose a novel quantum metrological scheme that utilizes a generating process with indefinite time direction to increase the canonical momentum nonlinearly. This approach achieves a nonlinear-scaling precision limit that improves quadratically with the increase in evolution time length or the number of processes, in accordance with Heisenberg's uncertainty relation. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed amount of energy in the probes. Consequently, our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on improving the precision of practical metrological and sensing tasks in realistic quantum systems.
Achieving the ultimate precisions for multiple parameters simultaneously is an outstanding challenge in quantum physics, because the optimal measurements for incompatible parameters cannot be performed jointly due to the Heisenberg uncertainty principle. In this work, a criterion proposed for multiparameter estimation provides a possible way to beat this curse. According to this criterion, it is possible to mitigate the influence of incompatibility meanwhile improve the ultimate precisions by increasing the variances of the parameter generators simultaneously. For demonstration, a scheme involving high-order Hermite-Gaussian states as probes is proposed for estimating the spatial displacement and angular tilt of light at the same time, and precisions up to 1.45 nm and 4.08 nrad are achieved in experiment simultaneously. Consequently, our findings provide a deeper insight into the role of Heisenberg uncertainty principle in multiparameter estimation, and contribute in several ways to the applications of quantum metrology.
Super-Heisenberg scaling, which scales as N − β with β > 1 in terms of the number of particles N or T − β in terms of the evolution time T , is better than Heisenberg scaling in quantum metrology. It has been proven that super-Heisenberg scaling can be achieved when the Hamiltonian of the system involves many-body interactions or time-dependent terms. We demonstrate that nonlinear quantum scrambling facilitates the achievement of super-Heisenberg scaling T − β when the generator of the parameter is time-independent. More importantly, in dissipative systems, we can still obtain super-Heisenberg scaling. In the optical cavity system, an exponential improvement in measurement precision over time can be achieved by combining injected external squeezing and intracavity squeezing. Our work provides an optimal method for leveraging nonlinear resources to enhance the measurement precision of the driving field.
The Heisenberg scaling is an ultimate precision limit of parameter estimation allowed by the principles of quantum mechanics, with no counterpart in the classical realm, and has been a long-pursued goal in quantum metrology. It has been known that interactions between the probes can help reach the Heisenberg scaling without entanglement. In this paper, we show that interactions between the probes and the additional dimensions of an ancillary system may also increase the precision of parameter estimation to surpass the standard quantum limit and attain the Heisenberg scaling without entanglement, if the measurement scheme is properly designed. The quantum Fisher information exhibits periodic patterns over the evolution time, implying the existence of optimal time points for measurements that can maximize the quantum Fisher information. By implementing optimizations over the Hamiltonian, the initial states of the probes and the ancillary system, the interaction strength, and the time points for measurements, our protocol achieves the Heisenberg scaling for the parameter of the probe Hamiltonian, in terms of both evolution time and probe number. Our protocol features two aspects: (i) the Heisenberg scaling can be achieved by a product state of the probes and (ii) mere local measurement on the ancilla is sufficient, both of which reduce the quantum resources and the implementation complexity to achieve the Heisenberg scaling. The paper is concluded by the investigation of the effects of noise on this protocol.