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arXiv:1201.2225 · Physical Review A

Ultimate limits to quantum metrology and the meaning of the Heisenberg limit

Marcin Zwierz, Carlos A. Pérez-Delgado, Pieter Kok

Published Apr 18, 2012• 78 citations
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Abstract

For the last 20 years, the question of what are the fundamental capabilities of quantum precision measurements has sparked a lively debate throughout the scientific community. Typically, the ultimate limits in quantum metrology are associated with the notion of the Heisenberg limit expressed in terms of the physical resources used in the measurement procedure. Over the years, a variety of different physical resources were introduced, leading to a confusion about the meaning of the Heisenberg limit. Here, we review the mainstream definitions of the relevant resources and introduce the universal resource count, that is, the expectation value of the generator (above its ground state) of translations in the parameter we wish to estimate, that applies to all measurement strategies. This leads to the ultimate formulation of the Heisenberg limit for quantum metrology. We prove that this limit holds for the generators of translations with an upper-bounded spectrum.

DOI
10.1103/physreva.85.042112

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For the last 20 years, the question of what are the fundamental capabilities of quantum precision measurements has sparked a lively debate throughout the scientific community. Typically, the ultimate limits in quantum metrology are associated with the notion of the Heisenberg limit expressed in terms of the physical resources used in the measurement procedure. Over the years, a variety of different physical resources were introduced, leading to a confusion about the meaning of the Heisenberg limit. Here, we review the mainstream definitions of the relevant resources and introduce the universal resource count, that is, the expectation value of the generator (above its ground state) of translations in the parameter we wish to estimate, that applies to all measurement strategies. This leads to the ultimate formulation of the Heisenberg limit for quantum metrology. We prove that this limit holds for the generators of translations with an upper-bounded spectrum.
Quantum metrology promises improved sensitivity in parameter estimation over classical procedures. However, there is a debate over the question of how the sensitivity scales with the resources and the number of queries that are used in estimation procedures. Here, we reconcile the physical definition of the relevant resources used in parameter estimation with the information-theoretical scaling in terms of the query complexity of a quantum network. This leads to a completely general optimality proof of the Heisenberg limit for quantum metrology. We give an example of how our proof resolves paradoxes that suggest sensitivities beyond the Heisenberg limit, and we show that the Heisenberg limit is an information-theoretic interpretation of the Margolus-Levitin bound, rather than Heisenberg's uncertainty relation.
Quantum metrology promises sensitivity beyond classical strategies, yet it remains unsettled how quantum-enabled precision should scale with physical resources and how to interpret quantum advantage. We provide a physically grounded resource accounting that clarifies the true Heisenberg limit and resolves apparent super-Heisenberg paradoxes. We demonstrate that the Heisenberg limit is best viewed as an information-theoretic manifestation of the quantum speed limit. We illustrate these ideas with a simple, super-resolving phase-estimation protocol based on Rabi oscillations in two-level atoms driven on an mmm-photon resonance. In this setting, the phase error scales as n−m/2n^{-m/2}n−m/2, where nnn is the average photon number. Recasting metrological sensitivity through quantum dynamical speed limits yields operational bounds that reconcile such super-resolution strategies with the standard Heisenberg interpretation and identify the relevant resources in the norm of the generator. We also revisit the common attribution of the NOON state's 1/n1/n1/n scaling to quantum entanglement. We show that such an attribution is not generic and the Heisenberg 1/n1/n1/n scaling does not, by itself, certify entanglement as the enabling resource.
Quantum metrology is a general term for methods to precisely estimate the value of an unknown parameter by actively using quantum resources. In particular, some classes of entangled states can be used to significantly suppress the estimation error. Here we derive a formula for rigorously evaluating an upper bound for the estimation error in a general setting of quantum metrology with arbitrary finite data sets. Unlike in the standard approach, where lower bounds for the error are evaluated in an ideal setting with almost infinite data, our method rigorously guarantees the estimation precision in realistic settings with finite data. We also prove that our upper bound shows the Heisenberg limit scaling whenever the linearized uncertainty, which is a popular benchmark in the standard approach, shows it. As an example, we apply our result to a Ramsey interferometer, and numerically show that the upper bound can exhibit the quantum enhancement of precision for finite data.
Quantum metrology aims to enhance measurement precision beyond the classical limit by leveraging quantum resources. Unlike multi-parameter dynamic quantum metrology, many questions regarding multiparameter quantum metrology at thermal equilibrium remain elusive. In particular, the ultimate precision limits achievable in this equilibrium setting are not yet well understood. In this work, we examine the fundamental limits of estimating multiple parameters with a quantum probe at thermal equilibrium. We first show that the Heisenberg limit with respect to the number of probes can be achieved, and our bound coincides with the known single-parameter bound when only one parameter is estimated. We then consider the low temperature limit, revealing a qualitatively different behavior compared to the finite temperature case. We give an example to illustrate the usage of our main results. Finally, we show the conditions under which the sensitivity bound can be attained and the optimal measurements to achieve it.
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.
This thesis presents three different results in quantum information theory. The first result addresses the theoretical foundations of quantum metrology. The Heisenberg limit considered as the ultimate limit in quantum metrology sets a lower bound on how precisely a physical quantity can be measured given a certain amount of resources in any possible measurement. Recently, however, several measurement procedures have been proposed in which the Heisenberg limit seemed to be surpassed. This led to an extensive debate over the question how the sensitivity scales with the physical resources and the computational resources that are used in estimation procedures. Here, we reconcile the physical definition of the relevant resources with the information-theoretical scaling in terms of the query complexity of a quantum network. This leads to a novel and ultimate Heisenberg limit that applies to all conceivable measurement procedures. The second result reveals a close relationship between quantum metrology and the Deutsch-Jozsa algorithm over continuous-variable quantum systems. Here, we develop a general procedure, characterized by two parameters, that unifies parameter estimation and the Deutsch-Jozsa algorithm. The procedure estimates a value of an unknown parameter with Heisenberg-limited precision or solves the Deutsch-Jozsa problem in a single run without the use of any entanglement. The third result illustrates how physical principles that govern interaction of light and matter can be efficiently employed to create a computational resource for a (one-way) quantum computer. More specifically, we demonstrate theoretically a scheme based on atomic ensembles and the dipole blockade mechanism for generation of the so-called cluster states in a single step. This procedure is significantly more efficient than any known robust probabilistic entangling operation.