Quantum Fisher Information Matrix Research Articles

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

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

Individual versus simultaneous estimation of two parameters in a correlated noisy environment

Recently, the phase parametric estimation for a single qubit system driven by a phase noisy laser under non-Markovian dynamics has been reported [Annals of Physics 351, 952 (2014)]. We here generalize the single-parameter estimation to the two-parameter scenario where a two-qubit system is influenced by the classical correlated noisy laser, and compare the performances of two-parameter estimation in both individual and simultaneous strategies by using the quantum Fisher information and the quantum Fisher information matrix, respectively. Our results show the phase parametric estimation precision can be significantly improved due to the memory effect arising from successive applications of the noisy laser. With the memory coefficient μ increasing, the precision of parameter estimation becomes more accurate. Besides, the phase parameter estimation precision can be drastically enhanced by engineering the ratios of classical phase diffusion rate and system-environment coupling strength in both Markovian and non-Markovian regions. Remarkably, we find the simultaneous estimation of two-phase parameters is not always advantageous over the individual strategy.

Optimal Generators for Quantum Sensing.

We propose a computationally efficient method to derive the unitary evolution that a quantum state is most sensitive to. This allows one to determine the optimal use of an entangled state for quantum sensing, even in complex systems where intuition from canonical squeezing examples breaks down. In this paper we show that the maximal obtainable sensitivity using a given quantum state is determined by the largest eigenvalue of the quantum Fisher information matrix (QFIM) and the corresponding evolution is uniquely determined by the coinciding eigenvector. Since we optimize the process of parameter encoding rather than focusing on state preparation protocols, our scheme is relevant for any quantum sensor. This procedure naturally optimizes multiparameter estimation by determining, through the eigenvectors of the QFIM, the maximal set of commuting observables with optimal sensitivity.

A stochastic evaluation of quantum Fisher information matrix with generic Hamiltonians

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.

Multiparameter estimation of quantum metrology in a Heisenberg system with Dzyaloshinskii–Moriya interaction

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.

Lee-Yang zeros and quantum Fisher information matrix in a nonlinear system.

The distribution of Lee-Yang zeros not only matters in thermodynamics and quantum mechanics, but also in mathematics. Hereby we propose a nonlinear quantum toy model and discuss the distribution of corresponding Lee-Yang zeros. Utilizing the coupling between a probe qubit and the nonlinear system, all Lee-Yang zeros can be detected in the dynamics of the probe qubit by tuning the coupling strength and linear coefficient of the nonlinear system. Moreover, the analytical expression of the quantum Fisher information matrix at the Lee-Yang zeros is provided and an interesting phenomenon is discovered. Both the coupling strength and temperature can simultaneously attain their precision limits at the Lee-Yang zeros. However, the probe qubit cannot work as a thermometer at a Lee-Yang zero if it sits on the unit circle.

Fisher information matrix as a resource measure in the resource theory of asymmetry with general connected-Lie-group symmetry

In recent years, in quantum information theory, there has been a remarkable development in the general theoretical framework for studying symmetry in dynamics. This development, called resource theory of asymmetry, is expected to have a wide range of applications, from accurate time measurements to black-hole physics. Despite its importance, the foundation of resource theory of asymmetry still has room for expansion. An important problem is in quantifying the amount of resource. When the symmetry prescribed U(1) or $\mathbb{R}$, i.e., with a single conserved quantity, the quantum Fisher information is known as a resource measure that has suitable properties and a clear physical meaning related to quantum fluctuation of the conserved quantity. However, it is not clear what is the resource measure with such suitable properties when a general symmetry prevails for which there are multiple conserved quantities. The purpose of this paper is to fill this gap. Specifically, we show that the quantum Fisher information matrix is a resource measure whenever a connected linear Lie group describes the symmetry. We also consider the physical meaning of this matrix and see which properties that the quantum Fisher information has when the symmetry is described by U(1) or $\mathbb{R}$ can be inherited by the quantum Fisher information matrix.

Simultaneous quantum estimation of phase and indistinguishability in a two-photon interferometer

With the rapid development of quantum technologies in recent years, the need for high-sensitivity measuring techniques has become a key issue. In particular, optical sensors based on quantum states of light have proven to be optimal resources for high-precision interferometry. Nevertheless, their performance may be severely affected by the presence of noise or imperfections. In this work, we derive the quantum Fisher information matrix associated with the simultaneous estimation of an interferometric phase and the indistinguishability characterizing the probe state consisting of an even number of photons. We find the optimal measurement attaining the ultimate precision for both parameters in a single setup and perform an experiment based on a pair of photons with an unknown degree of indistinguishability entering a two-port interferometer.

Sensing Magnetic Field with Single-Spin Dynamical Probe State: Control over Sensing Precision via Quantum Fisher Information

Quantum sensors play an important role in many branches of modern science, and they occupy a huge segment of the growing market for quantum devices. Quantum sensors use qubits and their analogs as detecting and analyzing quantum elements. Some sensors can be based on a single qubit, which is often presented as a system making its evolution on the so-called Bloch sphere. Different criteria are used to evaluate the efficiency of the sensing process. One of the most popular is the Quantum Fisher Information Matrix (QFIM) based on Fisher information. The magnitudes of the QFIM elements are strongly related to the precision of the sensing. As an analog of the classical Cramér theorem, one can define the quantum Cramér-Rao bound for the variance V, which is equal to V = 1/NF where F is the corresponding quantum Fisher information element, and N stands for the number of repeated sensory measurements. In this work, we develop our quantum Fisher information-based approach for a single feedback-driven qubit-type element for sensing external magnetic fields. We demonstrate the efficiency of our algorithm and discuss its further possible improvement. The approach developed here can be easily extended to other sensing schemes: collective spin systems and multi-qubit-based sensors. Alternative control algorithms can be applied to drive the probe state vector for maximization of the QFIM components. The particular choice of the control algorithm is defined by the specific experimental set-up.

Improving the Precision of Multiparameter Estimation in the Teleportation of Qutrit under Amplitude Damping Noise

AbstractSince the initial discovery of quantum teleportation, it is devoted to transferring unknown quantum states from one party to another distant partner. However, in the scenarios of remote sensing, what people truly care about is the information carried by certain parameters. The problem of multiparameter estimation in the framework of qutrit teleportation under amplitude damping (AD) noise is studied. Particularly, two schemes are proposed to battle against AD noise and enhance the precision of multiparameter estimation by utilizing weak measurement (WM) and environment‐assisted measurement (EAM). For two‐phase parameters encoded in a qutrit state, the analytical formulas of the quantum Fisher information matrix can be obtained. The results prove that the scheme of EAM outperforms the WM one in the improvements of both independent and simultaneous estimation precision. Remarkably, the EAM scheme can completely ensure the estimation precision against the contamination by AD noise. The reason should be attributed to the fact that EAM is carried out after the AD noise. Thus, it extracts information from both the system and the environment. The findings show that the techniques of WM and EAM are helpful for remote quantum sensing and can be generalized to other qutrit‐based quantum information tasks under AD decoherence.

Quantum Fisher information and parameter estimation in non-Hermitian Hamiltonians

Quantum Fisher information bounds the ultimate precision limit in the parameter estimation and has important applications in quantum metrology. In recent years, the theoretical and experimental studies of non-Hermitian Hamiltonians realized in quantum systems have attracted wide attention. Here, the parameter estimation based on eigenstates of non-Hermitian Hamiltonians is investigated, and the corresponding quantum Fisher information and quantum Cramér-Rao bound for the single-parameter and two-parameter estimations are given. In particular, the quantum Fisher information about estimating intrinsic momentum and external parameters in the non-reciprocal and gain-and-loss Su-Schrieffer-Heeger models, and non-Hermitian versions of the quantum Ising chain, Chern-insulator model and two-level system are calculated and analyzed. For these non-Hermitian models, the results show that in the case of single-parameter estimation in these non-Hermitian models, the quantum Fisher information increases significantly in the gapless regime and near the exceptional points, which can improve the accuracy limit of parameter estimation. For the two-parameter estimation, the determinant of the quantum Fisher information matrix also increases obviously near the gapless and exceptional points. In addition, a higher overall accuracy can be achieved in the topological regime than in the trivial regime, and the topological bound in two-parameter estimation can be determined by the Chern number.

Quantum Fisher information matrix for the two-axis twisting model at thermal equilibrium

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.

Unbounded and lossless compression of multiparameter quantum information

Several tasks in quantum-information processing involve quantum learning. For example, quantum sensing, quantum machine learning, and quantum-computer calibration involve learning and estimating unknown parameters $\mathbit{\ensuremath{\theta}}=({\ensuremath{\theta}}_{1},\ensuremath{\cdots},{\ensuremath{\theta}}_{M})$ from measurements of many copies of a quantum state ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}$. This type of metrological information is described by the quantum Fisher information matrix, which bounds the average amount of information learned about $\mathbit{\ensuremath{\theta}}$ per measurement of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}$. In several scenarios, it is advantageous to compress the multiparameter information encoded in ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\bigotimes}n}$ into ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}^{\text{ps}}\phantom{\rule{0.16em}{0ex}}{}^{\ensuremath{\bigotimes}m}$, where $m\ensuremath{\ll}n$. Here, we present a ``go-go'' theorem proving that $m/n$ can be made arbitrarily small, and that the information compression can happen without loss of information. We also demonstrate how to construct filters that perform this unbounded and lossless information compression. These filters can, for example, reduce arbitrarily the quantum-state intensity on experimental detectors, while retaining all initial information. Finally, we prove that the ability to compress quantum Fisher information is a nonclassical advantage that stems from the negativity of a particular quasiprobability distribution, a quantum extension of a probability distribution.

Quantum metrology with multimode Gaussian states of multiple point sources

A unified theoretical framework is developed for measuring multimode Gaussian states of multiple point sources. The spatial and temporal mode functions are found analytically by aid of the circular symmetry of the object plane and aperture, as well as by the rectangular spectral density function of the background. They relate the modal operators and statistical moments to the locations of the point sources through the prolate spheroidal functions. An experimentally straightforward implementation is proposed, which consists of a nonabsorptive polarization- and phase-modulating spatial light modulator (SLM), three lenses, and a polarization beam splitter. Interestingly, it is found that the spatial modal annihilation operators are related to the modal field operators in the image plane by a simple complex-valued scale factor. The quantum Fisher information matrix and quantum Cram\'er-Rao bound (QCRB) are also discussed. The numerical calculation of QCRB is completed for the case where the spatially multimode aperture fields are prepared by the spatial propagation from temporally single-mode thermal states, coherent states, or quadrature squeezed states of the point sources in the object plane. The results show that the Rayleigh limit can always be surpassed by means of increasing the mean photon rate or the squeeze parameter.

Multiparameter quantum metrology with discrete-time quantum walks

We address multi-parameter quantum estimation for one-dimensional discrete-time quantum walks and its applications to quantum metrology. We use the quantum walker as a probe for unknown parameters encoded on its coin degrees of freedom. We find an analytic expression of the quantum Fisher information matrix for the most general coin operator, and show that only two out of the three coin parameters can be accessed. We also prove that the resulting two-parameter coin model is asymptotically classical i.e. the Uhlmann curvature vanishes. Finally, we apply our findings to relevant case studies, including the simultaneous estimation of charge and mass in the discretized Dirac model.

Non-Markovian measure independent of initial states of open systems

In recent years, quantifying non-Markovian effect in open quantum system has become an important subject in the quantum decoherence control field. In this paper, a non-Markovian measure independent of the initial state of open system is proposed, thereby extending non-Markovian measure based on quantum Fisher information from the case where the initial state of the system is a pure state to the case where the initial state of the system is an arbitrary mixed state. As its application, the non-Markovian process is quantified by quantum Fisher information about a two-level system undergoing the three well-known dissipative channels, i.e. amplitude dissipative channel, phase damping channel, and random unitary channel. The results show that the conditions of non-Markovian processes in the three dissipative channels are independent of the selection of the initial state of the system by means of the quantum Fisher information of a phase parameter. Further, for amplitude dissipation channel and phase damping channel, the conditions for the non-Markovian processes to occur are equivalent to those given by trace distance, divisibility, quantum mutual information, quantum Fisher-information matrix, et al. As expected, for the case of amplitude dissipation channel, the corresponding results can reduce to the one in other paper (Lu X M, Wang X G, Sun C P <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1103/PhysRevA.82.042103">2010 <i>Phys. Rev. A</i><italic/> <b>82</b> 042103</ext-link>) by selecting the initial state of the system as an optimal pure state. However, for random unitary channel, the conditions of non-Markovian process are not equivalent to those for other measures. In addition, we also obtain an interesting relationship between quantum Fisher information and quantum coherence of the open system in the three dissipative channels, namely the square of quantum <inline-formula><tex-math id="M2">\begin{document}$l_1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20221053_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20221053_M2.png"/></alternatives></inline-formula> coherence for the evolved state of system is exactly equal to the quantum Fisher information of the phase parameter. In a word, the obtained results not only improve the application scope of using the quantum Fisher information to detect non-Markovian effects in open systems, but also further highlight its important role in quantum information processing.

Quantum Fisher information bounds on precision limits of circular dichroism

Circular dichroism (CD) is a widely used technique for investigating optically chiral molecules, especially for biomolecules. It is thus of great importance that these parameters be estimated precisely so that the molecules with desired functionalities can be designed. In order to surpass the limits of classical measurements, we need to probe the system with quantum light. We develop quantum Fisher information matrix (QFIM) for precision estimates of the circular dichroism and the optical rotary dispersion for a variety of input quantum states of light. The Cramer-Rao bounds, for all four chirality parameters are obtained, from QFIM for (a) single photon input states with a specific linear polarization and for (b) NOON states having two photons with both either left polarized or right polarized. The QFIM bounds, using quantum light, are compared with bounds obtained for classical light beams i.e., beams in coherent states. Quite generally, both the single photon state and the NOON state exhibit superior precision in the estimation of absorption and phase shift in relation to a coherent source of comparable intensity, especially in the weak absorption regime. In particular, the NOON state naturally offers the best precision among the three. We compare QFIM bounds with the error sensitivity bounds, as the latter are relatively easier to measure whereas the QFIM bounds require full state tomography. We also outline an empirical scheme for estimating the measurement sensitivities by projective measurements with single-photon detectors.

Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information

The Quantum Fisher Information matrix (QFIM) is a central metric in promising algorithms, such as Quantum Natural Gradient Descent and Variational Quantum Imaginary Time Evolution. Computing the full QFIM for a model withdparameters, however, is computationally expensive and generally requiresO(d2)function evaluations. To remedy these increasing costs in high-dimensional parameter spaces, we propose using simultaneous perturbation stochastic approximation techniques to approximate the QFIM at a constant cost. We present the resulting algorithm and successfully apply it to prepare Hamiltonian ground states and train Variational Quantum Boltzmann Machines.

Measurement Cost of Metric-Aware Variational Quantum Algorithms

We consider metric-aware quantum algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object. For example, the recently introduced quantum natural gradient approach uses the Fisher matrix as a metric tensor to correct the gradient vector for the codependence of the circuit parameters. We rigorously characterize and upper bound the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries. Finally, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.Received 18 May 2020Accepted 28 June 2021DOI:https://doi.org/10.1103/PRXQuantum.2.030324Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum algorithmsQuantum measurementsQuantum simulationQuantum InformationGeneral Physics

Quantum-limited localization and resolution in three dimensions

As a method to extract information from optical systems, imaging can be viewed as a parameter estimation problem. The fundamental precision in locating one emitter or estimating the separation between two incoherent emitters is bounded below by the multiparameter quantum Cramér-Rao bound (QCRB). Multiparameter QCRB gives an intrinsic bound in parameter estimation. We determine the ultimate potential of quantum-limited imaging for improving the resolution of a far-field, diffraction-limited optical field within the paraxial approximation. We show that the quantum Fisher information matrix (QFIm) in about one emitter’s position is independent on its true value. We calculate the QFIm of two unequal-brightness emitters’ relative positions and intensities; the results show that only when the relative intensity and centroids of two-point sources, including longitudinal and transverse directions, are known exactly, the separation in different directions can be estimated simultaneously with finite precision. Our results give the upper bounds on certain far-field imaging technology and will find wide use in applications from microscopy to astrometry.