Quantum Fisher Information Matrix (QFIM) is a fundamental concept in quantum-theoretical measurement that emphasizes the importance of quantum Cramér–Rao bound in estimating quantum parameters. Many relationships have been discovered between QFIM and many recent areas of research in quantum information. These relationships indicate that QFIM is not just a concept in quantum metrology but is a fundamental quantity in quantum information. The ability to estimate parameters and describe the optimal induced conditions for the thermal state of the two-axis twisting model was examined. QFIM is used as an estimator for multiple parameters. We show that simultaneous estimation reduces the estimate of parameter variation, so the accuracy of simultaneously estimating parameters is much better than individually. Each parameter can be used to control the criterion to reduce the minimum differences between other parameters.
Quantum Fisher Information Matrix (QFIM) is a fundamental concept in quantum-theoretical measurement that emphasizes the importance of quantum Cramér–Rao bound in estimating quantum parameters. Many relationships have been discovered between QFIM and many recent areas of research in quantum information. These relationships indicate that QFIM is not just a concept in quantum metrology but is a fundamental quantity in quantum information. The ability to estimate parameters and describe the optimal induced conditions for the thermal state of the two-axis twisting model was examined. QFIM is used as an estimator for multiple parameters. We show that simultaneous estimation reduces the estimate of parameter variation, so the accuracy of simultaneously estimating parameters is much better than individually. Each parameter can be used to control the criterion to reduce the minimum differences between other parameters.
Abstract Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cramér–Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cramér–Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.
We can utilize the Fisher quantum information matrix to evaluate the precision of any multiparametric forecasting methodology, due to the quantum Cramér–Rao bound. Using the density matrix vectorization approach, we examine both simultaneous and individual estimation procedures and the ratios of different estimation strategies. Our investigation takes into account interaction and intrinsic decoherence parameters, along with time, across various parameter combinations in a two-twisting axis model. Our findings suggest that using a simultaneous approach for multiple parameters in multiparameter estimation methods is generally more effective, offering higher precision than strategies that focus on individual parameters.
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.
We investigate multiparameter quantum estimation in a Raman-coupled two-qubit system at thermal equilibrium. Analytical expressions for the quantum Fisher information matrix are derived to characterize the simultaneous estimation of the temperature and Raman coupling strength. The corresponding quantum Cramér--Rao bounds are obtained and compared with those of individual estimation strategies. Our results reveal optimal operating regimes determined by the interplay between thermal fluctuations and coherent interactions. In particular, quantum thermometry exhibits a well-defined optimal temperature window, whereas the estimation of the Raman coupling strength is significantly enhanced in the low-temperature and weak-coupling regime. We further show that simultaneous estimation can outperform independent estimation within appropriate parameter regions, highlighting the advantages of multiparameter quantum metrology. These results provide analytical insights into the ultimate precision limits of Raman-coupled two-qubit systems and identify promising operating regimes for quantum sensing and quantum thermometry.
Abstract Quantum Fisher information matrix (QFIM) is a fundamental quantity in quantum physics, which closely links to diverse fields such as quantum metrology, phase transitions, entanglement witness, and quantum speed limit. It is crucial in quantum parameter estimation, central to the ultimate Cramér-Rao bound. Recently, the evaluation of QFIM using quantum circuit algorithms has been proposed for systems with multiplicative parameters Hamiltonian. However, systems with generic Hamiltonians still lack these proposed schemes. This work introduces a quantum-circuit-based approach for evaluating QFIM with generic Hamiltonians. We present a time-dependent stochastic parameter-shift rule for the derivatives of evolved quantum states, whereby the QFIM can be obtained. The scheme can be executed in universal quantum computers under the family of parameterized gates. In magnetic field estimations, we demonstrate the consistency between the results obtained from the stochastic parameter-shift rule and the exact results, while the results obtained from a standard parameter-shift rule slightly deviate from the exact ones. Our work sheds new light on studying QFIM with generic Hamiltonians using quantum circuit algorithms.
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.