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arXiv:1409.2200 · Physical Review A

Multiple phase estimation for arbitrary pure states under white noise

Yao Yao, Li Ge, Xing Mao Xiao, Xiaoguang Wang, C. P. Sun

Published Dec 8, 2014• 42 citations
arXivHTMLPDF

Abstract

In any realistic quantum metrology scenarios, the ultimate precision in the estimation of parameters is limited not only by the so-called Heisenberg scaling, but also the environmental noise encountered by the underlying system. In the context of quantum estimation theory, it is of great significance to carefully evaluate the impact of a specific type of noise on the corresponding quantum Fisher information or quantum Fisher information matrix (QFIM). Here we investigate the multiple phase estimation problem for a natural parametrization of arbitrary pure states under white noise. We obtain the explicit expression of the symmetric logarithmic derivative (SLD) and hence the analytical formula of QFIM. Moreover, the attainability of the quantum Cram\'er-Rao bound is confirmed by the commutability of SLDs and the optimal estimators are elucidated for experimental purposes. These findings generalize previously known partial results and highlight the role of white noise in quantum metrology.

DOI
10.1103/physreva.90.062113

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Abstract We derive several expressions for the quantum Fisher information matrix (QFIM) for the multi-parameter estimation of multi-mode Gaussian quantum states, the corresponding symmetric logarithmic derivatives, and conditions for saturability of the quantum Cramér–Rao bound. This bound determines the ultimate precision with which parameters encoded into quantum states can be estimated. We include expressions for mixed states, for the case when the Williamson decomposition of the covariance matrix is known, expressions in terms of infinite series, and expressions for pure states. We also discuss problematic behavior when some modes are pure, and present a method that allows the use of expressions that are defined only for mixed states, to compute QFIM for states with any number of pure modes.
The quantum Fisher information matrix (QFIM) is central to multiparameter quantum metrology, dictating the attainable sensitivity via the quantum Cramér-Rao bound. In this work, we investigate the ultimate precision limits for relativistic quantum thermometry in a bipartite system of uniformly accelerated Unruh-DeWitt detectors. Utilizing the symmetric logarithmic derivative (SLD) formalism within the QFIM framework, we analyze the individual and simultaneous estimation of the Unruh temperature TTT and the initial-state parameter Δ0Δ_0Δ0​. In the noiseless case, we demonstrate that these two parameters are quantum compatible, allowing the multiparameter quantum Cramér-Rao bound to be saturated without a loss of precision. We then examine the impact of environmental effects by comparing Markovian and non-Markovian dynamics. In the Markovian regime, dissipation leads to a monotonic degradation of estimation precision; conversely, non-Markovian memory effects induce temporal oscillations and transient precision enhancements due to information backflow. The robustness of the estimation protocols is further analyzed under correlated noisy channels, including amplitude damping, phase flip, and phase damping. We show that dissipative noise results in more significant precision loss than purely dephasing mechanisms, whereas classical correlations in the noise mitigate this degradation. Finally, we discuss the estimation of the detector energy-level spacing ωωω as a natural extension, highlighting its sensitivity to the environmental structure. Our results provide a unified framework for relativistic multiparameter quantum metrology within the context of open quantum systems.
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
Abstract Multiparameter quantum estimation theory aims to determine simultaneously the ultimate precision of all parameters contained in the state of a given quantum system. Determining this ultimate precision depends on the quantum Fisher information matrix (QFIM) which is essential to obtaining the quantum Cramér–Rao bound. This is the main motivation of this work which concerns the computation of the analytical expression of the QFIM. Inspired by the results reported in J. Phys. A: Math. Theor. 52 035304 (2019), the general formalism of the multiparameter quantum estimation theory of quantum Gaussian states in terms of their first and second moments is given. We give the analytical formulas of right logarithmic derivative and symmetric logarithmic derivative operators. Then we derive the general expressions of the corresponding quantum Fisher information matrices. We also derive an explicit expression of the condition which ensures the saturation of the quantum Cramér–Rao bound in estimating several parameters. Finally, we examine some examples to clarify the use of our results.
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Quantum metrology faces major challenges in noisy environments, where decoherence rapidly degrades useful quantum resources. We investigate the dynamics of the precision limits given by the quantum Fisher information (QFI) for phase estimation under spatially correlated dephasing. We characterize the dynamics of the QFI by the sensitivity and degradation indicators that can be obtained as analytical expressions derived using perturbative theory treatment. These short-time and weak-noise formulas yield analytic insight into how collective-spin moments govern both (i) the noiseless sensitivity and (ii) the leading noise-induced degradation of metrological usefulness. We identify a trade-off that is intrinsic to our commuting encoding-noise structure. We analyze the QFI dynamics for the Gaussian spin state (GSS) superpositions, encompassing spin coherent state (SCS), Dicke state superpositions, spin-squeezed states, and GHZ-like states. Predictions from indicators of the QFI dynamics are compared to both the quantum Cramér--Rao bound and the measurement-specific sensitivity bounds for an optimal parameter and interrogation time under a finite total time resource. When possible, we analytically derive the measurement-specific sensitivity bounds for spin-projection and parity-based measurements.