Significance: The experimental detection and theoretical characterization of entanglement in many-body systems is a mostly unsolved problem. Here we relate entanglement to the statistical speed measuring how quickly nearby states become distinguishable under the action of an arbitrary many-body Hamiltonian. The method is remarkably simple: We witness multipartite entanglement, just elaborating on published experimental data with ions and photons. The unveiled connection between the statistical speed and entanglement provides a unifying framework that can shed new light on quantum information science and quantum critical phenomena.

Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of an unknown parameter. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cramér–Rao bound, for instance, do not apply to Bayesian strategies and vice versa. In particular, we show that the Bayesian variance can overcome the frequentist Cramér–Rao bound, which appears to be a paradoxical result if the conceptual difference between the two approaches are overlooked. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.

We study the performance of an $N$-qubit $W$ superposition state composed of a $W$ state and its obverse in quantum metrology. Taking advantage of the general Ising-type Hamiltonian (including noninteracting and interacting operation), we analytically present the quantum Fisher information (QFI) of an $N$-qubit $W$ superposition state under different situations and then investigate its phase sensitivity. The results show that the phase sensitivity under noninteracting operation displays a crossover from the $W$ state to Greenberger-Horne-Zeilinger (GHZ) state, where it is same as $W$ state in the few-qubit case $(N\le 6)$ but asymptotically equal to the GHZ state for large-qubit cases $(N\gg 1)$. Interestingly, the 4-qubit $W$ superposition state is found to have the same sensitivity as the 4-qubit GHZ state. And the optimal measurement protocols are provided for ideal metrology. Under the phase-amplitude damping channel, the phase sensitivity of the $W$ superposition state (except for $N=3$) is ultimately decreased to the standard quantum limit, while it turns worse in a depolarizing channel. Finally, the tunable phase sensitivity under interacting operation is studied, and the general Heisenberg limit is surpassed with the increasing interaction strength $\gamma$. Meanwhile, a plateau of QFI and phase sensitivity is found for all large-qubit $W$ superposition states, which is similar to the study of the GHZ state and again verifies the common feature of GHZ-type states in quantum metrology.

Sensitivity bounds for a generic interferometric phase estimation problem are the shot noise and the Heisenberg limits. The shot noise is the highest sensitivity that can be reached with separable states, while the Heisenberg limit is the ultimate bound in sensitivity which can be saturated with entangled states. The scaling of these bounds with the number of particles $N$ entering the interferometer depends on the specific Hamiltonian governing the same interferometer. In typical cases, the Hamiltonian is linear, and the shot-noise and Heisenberg limits scale with $N^{-1/2}$ and with $N^{-1}$, respectively. With interferometers described by generic, nonlinear Hamiltonian, the scalings with the number of particles can be rather different. Here we study the shot-noise and Heisenberg limits of Ising-like Hamiltonians in the presence of longitudinal and transverse fields, both in the nearest-neighbor and in the fully connected spin interaction cases. We provide the explicit forms of the states saturating the shot-noise and Heisenberg limits. These results can be relevant not only for precision measurement purposes but also to characterize quantum phase transitions and, more generally, the witnessing of multipartite entanglement.

Quantum metrology is one of the hot topics in ultra-cold atoms physics. It is now well established that with the help of entanglement, the measurement sensitivity can be greatly improved with respect to the current generation of interferometers that are using classical sources of particles. Recently, Quantum Fisher information plays an important role in this field. In this paper, a brief introduction on Quantum metrology is presented highlighting the role of the Quantum Fisher information. And then a brief review on the recent developments for i) criteria of multi-particle entanglement and its experimental generation; ii) linear and non-linear atomic interferometers; iii) the effective statistical methods for the analysis of the experimental data.