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arXiv:1506.08837 · Physical Review A

Asymptotic role of entanglement in quantum metrology

Remigiusz Augusiak, Jan Kołodyński, Alexander Streltsov, Manabendra Nath Bera, Antonio Acín, Maciej Lewenstein

Published Jul 22, 2016• 45 citations
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Abstract

Quantum systems allow one to sense physical parameters beyond the reach of classical statistics---with resolutions greater than $1/N$, where $N$ is the number of constituent particles independently probing a parameter. In the canonical phase-sensing scenario the Heisenberg limit $1/{N}^{2}$ may be reached, which requires, as we show, both the relative size of the largest entangled block and the geometric measure of entanglement to be nonvanishing as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Yet, we also demonstrate that in the asymptotic $N$ limit any precision scaling arbitrarily close to the Heisenberg limit ($1/{N}^{2\ensuremath{-}\ensuremath{\varepsilon}}$ with any $\ensuremath{\varepsilon}>0$) may be attained, even though the system gradually becomes noisier and separable, so that both the above entanglement quantifiers asymptotically vanish. Our work shows that sufficiently large quantum systems achieve nearly optimal resolutions despite their relative amount of entanglement being arbitrarily small. In deriving our results, we establish the continuity relation of the quantum Fisher information evaluated for a phaselike parameter, which lets us link it directly to the geometry of quantum states, and hence naturally to the geometric measure of entanglement.

DOI
10.1103/physreva.94.012339

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Quantum systems allow one to sense physical parameters beyond the reach of classical statistics---with resolutions greater than 1/N1/N1/N, where NNN is the number of constituent particles independently probing a parameter. In the canonical phase-sensing scenario the Heisenberg limit 1/N21/{N}^{2}1/N2 may be reached, which requires, as we show, both the relative size of the largest entangled block and the geometric measure of entanglement to be nonvanishing as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. Yet, we also demonstrate that in the asymptotic NNN limit any precision scaling arbitrarily close to the Heisenberg limit (1/N2\ensuremath−\ensuremathε1/{N}^{2\ensuremath{-}\ensuremath{\varepsilon}}1/N2\ensuremath−\ensuremathε with any \ensuremathε>0\ensuremath{\varepsilon}>0\ensuremathε>0) may be attained, even though the system gradually becomes noisier and separable, so that both the above entanglement quantifiers asymptotically vanish. Our work shows that sufficiently large quantum systems achieve nearly optimal resolutions despite their relative amount of entanglement being arbitrarily small. In deriving our results, we establish the continuity relation of the quantum Fisher information evaluated for a phaselike parameter, which lets us link it directly to the geometry of quantum states, and hence naturally to the geometric measure of entanglement.
Quantum systems allow one to sense physical parameters beyond the reach of classical statistics---with resolutions greater than 1/N1/N1/N, where NNN is the number of constituent particles independently probing the parameter. In principle, the Heisenberg Limit 1/N21/N^{2}1/N2 may even be reached, which requires both the relative size of the largest entangled block and the geometric measure of entanglement to be non-vanishing as N ⁣→ ⁣∞N\!\to\!\inftyN→∞. Yet, we demonstrate that in the asymptotic NNN limit any precision scaling arbitrarily close to the Heisenberg Limit (1/N2−ε1/N^{2-\varepsilon}1/N2−ε with any ε ⁣> ⁣0\varepsilon\!>\!0ε>0) may be attained, even though the system gradually becomes noisier and separable, and both the above entanglement quantifiers asymptotically vanish. Our results thus show that quantum systems achieve nearly optimal resolution scaling despite arbitrarily small amount of entanglement present. In deriving these results, we also establish a continuity relation for the Quantum Fisher Information of independent interest.
In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as 1/N21/N^21/N2 with the number of particles involved---substantially surpassing the 1/N1/N1/N-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic NNN limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large NNN the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.
It is a specific type of quantum correlated state that achieves optimal precision in parameterestimation under unitary encoding. We consider the potential experimental limitation on probe entanglement, and find a relation between achievable precision and initial probe entanglement, in both bipartite and multipartite scenarios. For two-qubit probes, we analytically derive an exact relationship between the entanglement-constrained optimal quantum Fisher information and the limited initial entanglement, measured via both generalized geometric measure and entanglement entropy. We demonstrate that this fundamental relationship persists across the same range of the entanglement measures even when higher-dimensional bipartite probes are considered. Furthermore, we identify the specific states that realize maximum precision in these scenarios. Additionally, by considering the geometric measure of entanglement, we extend our approach to multiqubit probes. We find that in every case, the optimal quantum Fisher information exhibits a universal behavior:a steep increase in the low-entanglement regime, followed by a gradual and nearly-saturated improvement as the probe entanglement approaches values close to those required for achieving the Heisenberg limit.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of NNN resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with NNN better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources NNN is strictly constrained, I show that the final measurement uncertainty is πππ times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.
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 address the framework of analysing quantum metrology in the information-theoretic picture. Firstly we show how to extract the maximum amount of information in general via suitable state initialization of the probes at the beginning and a quantum measurement at the end. Our analysis can apply to both the single-parameter and the multi-parameter estimation procedures as well as to any other quantum information processing procedures. We then establish a direct connection between the information-theoretic picture of quantum metrology and its conventional variance-covariance picture, by showing that any estimation procedure achieves Heisenberg limit in variance-covariance picture can also reach the information-theoretic Heisenberg limit in the asymptotic sense. As a direct consequence, we argue that the entangled measurement is not necessary for achieving Heisenberg limit in the information-theoretic pictures, which is explicitly illustrated for the Quantum-Classical parallel strategy of quantum metrology with a separable measurement employed and the Heisenberg limit saturated in both pictures.