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arXiv:1509.03334 · Physical Review Letters

Detecting Large Quantum Fisher Information with Finite Measurement Precision

Florian Fröwis, Pavel Sekatski, Wolfgang Dür

Published Mar 3, 2016• 146 citations
arXivHTMLPDF

Abstract

We propose an experimentally accessible scheme to determine the lower bounds on the quantum Fisher information (QFI), which ascertains multipartite entanglement or usefulness for quantum metrology. The scheme is based on comparing the measurement statistics of a state before and after a small unitary rotation. We argue that, in general, the limited resolution of collective observables prevents the detection of large QFI. This can be overcome by performing an additional operation prior to the measurement. We illustrate the power of this protocol for present-day spin-squeezing experiments, where the same operation used for the preparation of the initial spin-squeezed state improves also the measurement precision and hence the lower bound on the QFI by 2 orders of magnitude. We also establish a connection to the Leggett-Garg inequalities. We show how to simulate a variant of the inequalities with our protocol and demonstrate that large QFI is necessary for their violation with coarse-grained detectors.

DOI
10.1103/physrevlett.116.090801

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We propose an experimentally accessible scheme to determine the lower bounds on the quantum Fisher information (QFI), which ascertains multipartite entanglement or usefulness for quantum metrology. The scheme is based on comparing the measurement statistics of a state before and after a small unitary rotation. We argue that, in general, the limited resolution of collective observables prevents the detection of large QFI. This can be overcome by performing an additional operation prior to the measurement. We illustrate the power of this protocol for present-day spin-squeezing experiments, where the same operation used for the preparation of the initial spin-squeezed state improves also the measurement precision and hence the lower bound on the QFI by 2 orders of magnitude. We also establish a connection to the Leggett-Garg inequalities. We show how to simulate a variant of the inequalities with our protocol and demonstrate that large QFI is necessary for their violation with coarse-grained detectors.
The quantum Fisher information (QFI) of certain multipartite entangled quantum states is larger than what is reachable by separable states, providing a metrological advantage. Are these nonclassical correlations strong enough to potentially violate a Bell inequality? Here, we present evidence from two examples. First, we discuss a Bell inequality designed for spin-squeezed states which is violated only by quantum states with a large QFI. Second, we relax a well-known lower bound on the QFI to find the Mermin Bell inequality as a special case. However, a fully general link between QFI and Bell correlations is still open.
We prove that a violation of a Leggett-Garg inequality for bounded observables in stationary pure states and thermal states yields a rigorous lower bound on the quantum Fisher information. This turns a qualitative foundations test of realism in quantum systems into a quantitative witness of useful quantum sensitivity and, in the collective setting, into a lower bound on multipartite entanglement depth in many-body systems. We further demonstrate that Leggett-Garg violations are constrained by the same spectral moments, susceptibilities, and fff-sum-rule bounds that organize many-body response. Our results show that temporal correlations of a single collective observable can serve as an experimentally accessible witness of many-body quantum coherence, without requiring full state reconstruction.
Quantum-enhanced sensing is commonly benchmarked using the quantum Fisher information (QFI), often interpreted as a direct indicator of achievable precision. However, such an interpreta- tion is only justified within a fully specified inference framework that consistently incorporates state preparation, measurement design, resource accounting, estimator construction, prior information, and finite data effects. Here we develop an operational, end-to-end framework for quantum sensing under finite resources and identify general principles required for meaningful performance assess- ment. A central conceptual point is that the relevant unit of estimation is not a single detection event, but the complete data set required to construct a consistent estimator. Rather than relying on abstract statistical formalism, we analyze a set of paradigmatic and experimentally relevant ex- amples that expose common pitfalls in widely discussed quantum sensing strategies. These include quantum interferometry, sensing with squeezed states, and criticality-inspired protocols. Our results clarify when large Fisher information or nonclassical resources translate into genuine metrological advantage, and when they do not. They further provide a practical methodology for designing and evaluating quantum sensing protocols under realistic experimental constraints.
The quantum Fisher information (QFI) represents a fundamental concept in quantum physics. It quantifies the metrological potential of quantum states in quantum parameter estimation measurements, and is intrinsically related to quantum geometry and multipartite entanglement of many-body systems. Using a nitrogen-vacancy center spin in diamond, we experimentally demonstrate a randomized-measurement method to extract the QFI of the qubit, for both pure and mixed states. We then apply this scheme to a 4-qubit state, using a superconducting quantum computer, and show that it provides access to the sub-QFI, which sets a lower bound on the QFI for general mixed states. We numerically study the scaling of statistical error, considering NNN-qubit states, to illustrate the advantage of our randomized-measurement approach in estimating the QFI and multipartite entanglement. Our results highlight the general applicability of our method to different quantum platforms, including solid-state spin systems, superconducting quantum computers, and trapped ions.
Quantum Fisher information (QFI) is a fundamental quantifier in quantum metrology, determining the ultimate precision achievable in parameter-estimation protocols through the quantum Cramér-Rao bound. However, direct evaluation of the QFI generally requires detailed knowledge of the density matrix, making it increasingly demanding as the Hilbert-space dimension grows. In this work, we investigate the extent to which the QFI of multipartite quantum systems can be predicted from a limited set of experimentally accessible quantities using support vector regression (SVR). By comparing different physically motivated features, we identify a dominant feature set governing QFI and show that the predictive power of collective spin moments alone decreases as system size and consequently Hilbert-space dimension grows. We demonstrate that QFI is governed primarily by the interplay between collective covariance and low-order spectral moments of the density matrix. Our results identify the physically relevant information sectors governing the QFI and demonstrate that accurate estimation of metrological sensitivity can be achieved from a restricted set of experimentally accessible quantities without requiring full quantum-state tomography.
Quantum metrology exploits quantum correlations to make precise measurements with limited particle numbers. By utilizing inter- and intramode correlations in an optical interferometer, we find a state that combines entanglement and squeezing to give a sevenfold enhancement in the quantum Fisher information (QFI)---a metric related to the precision---over the shot-noise limit, for low photon numbers. Motivated by practicality we then look at the squeezed cat state, which has recently been made experimentally, and shows further precision gains over the shot-noise limit and a threefold improvement in the QFI over the optimal Gaussian state. We present a conceptually simple measurement scheme that saturates the QFI, and we demonstrate a robustness to loss for small photon numbers. The squeezed cat state can therefore give a significant precision enhancement in optical quantum metrology in practical and realistic conditions.