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arXiv:1407.8065 · Physical Review A

Precision-guaranteed quantum metrology

Takanori Sugiyama

Published Apr 27, 2015• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.91.042126

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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 precision enhancement is of fundamental importance for the development of advanced metrological optical experiments, such as gravitational wave detection and frequency calibration with atomic clocks. Precision in these experiments is strongly limited by the 1/√N shot noise factor with N being the number of probes (photons, atoms) employed in the experiment. Quantum theory provides tools to overcome the bound by using entangled probes. In an idealized scenario this gives rise to the Heisenberg scaling of precision 1/N. Here we show that when decoherence is taken into account, the maximal possible quantum enhancement in the asymptotic limit of infinite N amounts generically to a constant factor rather than quadratic improvement. We provide efficient and intuitive tools for deriving the bounds based on the geometry of quantum channels and semi-definite programming. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: depolarization, dephasing, spontaneous emission and photon loss. Quantum metrology employs the properties of quantum states to further enhance the accuracy of some of the most precise measurement schemes to date. Here, a method for estimating the upper bounds to achievable precision in quantum-enhanced metrology protocols in the presence of decoherence is presented.
Quantum state tomography is currently the standard tool for verifying that a state prepared in the lab is close to an ideal target state, but up to now there have been no rigorous methods for evaluating the precision of the state preparation in tomographic experiments. We propose a new estimator for quantum state tomography, and prove that the (always physical) estimates will be close to the true prepared state with a high probability. We derive an explicit formula for evaluating how high the probability is for an arbitrary finite-dimensional system and explicitly give the one- and two-qubit cases as examples. This formula applies for any informationally complete sets of measurements, arbitrary finite number of data sets, and general loss functions including the infidelity, the Hilbert-Schmidt, and the trace distances. Using the formula, we can evaluate not only the difference between the estimated and prepared states, but also the difference between the prepared and target states. This is the first result directly applicable to the problem of evaluating the precision of estimation and preparation in quantum tomographic experiments.
Quantum metrology calculates the ultimate precision of all estimation strategies, measuring what is their root-mean-square error (RMSE) and their Fisher information. Here, instead, we ask how many bits of the parameter we can recover; namely, we derive an information-theoretic quantum metrology. In this setting, we redefine "Heisenberg bound" and "standard quantum limit" (the usual benchmarks in the quantum estimation theory) and show that the former can be attained only by sequential strategies or parallel strategies that employ entanglement among probes, whereas parallel-separable strategies are limited by the latter. We highlight the differences between this setting and the RMSE-based one.
Quantum metrology offers the possibility of quantum enhancements of the precision of various sensing tasks. In this manuscript, we tackle two open problems in the theory of single-shot quantum parameter learning, going beyond the usual setting of local parameter estimation via repeated measurements. The first concerns the construction of global upper bounds on the learning precision. The second concerns rigorous guarantees on the success probability of parameter learning, namely, lower bounds on the probability of learning a parameter with a certain precision, given the constraints on the resources used for the quantum metrology task. We provide rigorous, practical, and global upper bounds and success-probability guarantees for quantum parameter learning. Most importantly, we establish a fidelity-based learning guarantee for generic mixed-state models that can be viewed as the achievability-side analogue of the quantum Cramer-Rao bound. Whereas the latter provides a no-go constraint, based on the local curvature of the fidelities, our bound uses only pairwise fidelities between parameter-encoded states to certify that a prescribed precision is attainable with a guaranteed success probability. We demonstrate the versatility of the new bounds in a Rabi-frequency-learning example involving a driven qubit coupled to a bosonic environment and a collective-spin Hamiltonian learning problem. Together, the new global bounds and success-probability guarantees allow us to rule out unattainable precision and to certify attainable precision beyond what is possible via standard Fisher-information analysis or binary hypothesis testing bounds. They also allow one to tractably characterize the performance of various learning schemes, without the overhead of an explicit simulation.
Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-12\frac{1}{2}21​ particle and compare the results for estimating the amplitude and the direction of the magnetic field.
We derive an explicit condition that determines whether in a noisy quantum frequency estimation problem the estimation precision of the most general adaptive quantum metrological protocol cannot reach the Heisenberg-like scaling. The condition is a simple algebraic statement on a relation between the Hamiltonian operator representing the unitary part of the dynamics and the noise operators appearing in the quantum Master equation, and does not require any finite-time integration of the dynamics. In particular these results allow us to understand when application of quantum error correction protocols in order to recover the Heisenberg scaling in quantum metrology is not possible. Additionally, we provide methods to obtain quantitative bounds on achievable precision in the most general adaptive quantum metrological models. Finally, we apply the newly developed tools to prove fundamental bounds in atomic interferometry with many-body effects such as many body losses as well as models involving many-body terms in the Hamiltonian part of the dynamics commonly referred to as non-linear quantum metrology.