Efficient qubit phase estimation using adaptive measurements

Abstract

Estimating correctly the quantum phase of a physical system is a central problem in quantum parameter estimation theory due to its wide range of applications from quantum metrology to cryptography. Ideally, the optimal quantum estimator is given by the so-called quantum Cramér-Rao bound, so any measurement strategy aims to obtain estimations as close as possible to it. However, more often than not, the current state-of-the-art methods to estimate quantum phases fail to reach this bound as they rely on maximum likelihood estimators of non-identifiable likelihood functions. In this work we thoroughly review various schemes for estimating the phase of a qubit, identifying the underlying problem which prohibits these methods to reach the quantum Cramér-Rao bound, and propose a new adaptive scheme based on covariant measurements to circumvent this problem. Our findings are carefully checked by Monte Carlo simulations, showing that the method we propose is both mathematically and experimentally more realistic and more efficient than the methods currently available.

Determination of the asymptotic limits of adaptive photon counting measurements for coherent-state optical phase estimation

“…These strategies maximize either the estimation precision (by minimizing the Holevo variance 51 ), or the information gain about the unknown parameter based on entropy measures, including mutual information, the Kullback-Lieber divergence, and the conditional entropy 53,54 . We note that these cost functions produce non-identifiable likelihood functions that do not allow to correctly estimate a cyclic parameter, such as the phase 55 . To address this problem, these non-Gaussian strategies use the Fisher information to optimize the displacement operations, which are the dynamical control variable in the strategy, to guarantee that these cost functions provide identifiable likelihood functions, and to enable optical phase estimation with near-optimal performance.…”

Efficient qubit phase estimation using adaptive measurements

“…Appendix A: Code implementation Our numerical results have been implemented using R language. An R library with the various methods discussed here can be publicly found in the repository [35]. To reproduce the numerical points for covariant strategy, AQSE and restricted-AQSE presented in Fig.…”

Estimação Bayesiana para termometria colisional e computação quântica não-holonômica em tempo ótimo

“…This example illustrates that it is not uncommon for the optimal basis to depend on the unknown parameter itself, introducing another layer of complexity to the problem. This often calls for adaptative strategies whenever possible, where one continuously change the measurement basis, or, e.g., parameters and the general configuration of the interaction [165][166][167]. a…”

“…Still, it is also important to remember that even MLE strategies might contain some limitations. This is a hurdle encountered in, e.g., phase estimation [166]. These considerations also highlight the importance of the FI and the QFI: even if we are either unable to obtain an optimal estimator (or to employ the best measurement strategy), the (quantum) Fisher information still provides an important benchmark.…”