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Quantum Studies Mathematics and Foundations

Quantum Fisher information matrix of quantum metrology in a Heisenberg XXZ model

Rachid Ben hammou, Abdelfattah El Achab, Nabil Habiballah

Published Feb 8, 2024• 0 citations
Publisher
DOI
10.1007/s40509-024-00315-w

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In this work, we study the quantum Fisher information on the Heisenberg model in presence of the Dzyaloshinskii–Moriya interaction, basing on the multi-parametric estimation strategy in quantum metrology. First, we employ the density matrix vectorization method with a tool of the quantum Fisher information matrix through quantum Cramer–Rao bound. Second, we show that the multi-parameter quantum Cramer–Rao bound for simultaneous and individual estimation of the temperature and Dzyaloshinskii–Moriya interaction strongly relies on the concepts of the quantum Fisher information matrix. We confirm also the simultaneous strategy that provides a high precision than the individual.
Recently, distributed sensing has been extensively studied using squeezed states. While this is an excellent development, it is desirable to investigate the use of other quantum probes, such as entangled states of light. In this paper, we focus on distributed sensing, i.e., estimating multiple unknown phases at different spatial nodes using multiphoton polarization-entangled Greenberger-Horne-Zeilinger states distributed across different nodes. We utilize tools of quantum metrology and calculate the quantum Fisher information matrix (QFIM). However, the QFIM turns out to be singular, hindering the determination of quantum Cramér-Rao bounds for the parameters of interest. Recent experiments have contended with a weaker form of the Cramér-Rao bound, which does not require the inversion of the QFIM. It is desirable to understand how relevant these weaker bounds are and how closely they approach the exact Cramér-Rao bounds. We thus analyze the reason for this singularity and, by removing a redundant phase, obtain a nonsingular QFIM, allowing us to derive exact quantum Cramér-Rao bounds. Using the nonsingular QFIM, we show that the arithmetic average of the distributed phases is Heisenberg limited. We demonstrate that quantum metrological bounds can be saturated by projective measurements, enabling us to determine the Fisher information matrix, which is also singular. We then show how this singularity can be resolved.
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.