Quantum metrology from a quantum information science perspective

Abstract

We summarise important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger-Horne-Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér-Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.

Frequentist and Bayesian Quantum Phase Estimation

“…In Equation (19), the prior distribution p pri (θ) expresses the a priori state of knowledge on θ, p(µ|θ) is the likelihood that is determined by the quantum mechanical measurement postulate, e.g., as in Equation (1), and the marginal probability p mar (µ) = b a dθ p(θ, µ) is obtained through the normalization for the posterior, where a and b are boundaries of the phase domain. The posterior probability p post (θ|µ) describes the current knowledge about the random variable θ based on the available information, i.e., the measurement results µ.…”

“…Using classically-correlated probe states, it is possible to reach the so-called shot noise or standard quantum limit, which is the limiting factor for the current generation of interferometers and sensors [9][10][11][12]. Strategies involving probe states characterized by squeezed quadratures [13] or entanglement between particles [14][15][16][17][18][19] are able to overcome the shot noise, the ultimate quantum bound being the so-called Heisenberg limit. Quantum noise reduction in phase estimation has been demonstrated in several proof-of-principle experiments with atoms and photons [20,21].…”

Witnessing Multipartite Entanglement by Detecting Asymmetry

“…There are several entanglement witnesses written in terms of the quantum Fisher information. They relate entanglement to the system speed of response to phase shifts generated by additive spin-1/2 Hamiltonians J N = ∑ N i=1 1/2σ i [32,46,[54][55][56][57][58][59][60]. In particular, a constraint which cannot be satisfied by k-separable states of N qubits is F J N (ρ) ≥ nk 2 + (N − nk) 2 , where n = N k .…”

“…The results are given in Table 1. An interesting alternative option is to build entanglement witnesses in terms of the average values and the variances of the collective spin operators (see [32] and references therein, and a newer proposal in [61]). The advantage of such methods is that the witness can be calculated without actually performing the phase shift, just by measuring the spin values on the probe state.…”

Metrological Measures of Non-classical Correlations

“…For enjoyable reviews on quantum metrology, we refer the Reader to Refs. [15,16]. It is indeed possible to take advantage of quantumness to increase the precision of measurement schemes.…”