Geometric bifurcation theory: Fisher information geometry applied to dynamical and complex systems

The prediction of material lifetime is central to nanomaterial engineering and reliability analysis. We propose the Marshall–Olkin Power Half-Logistic (MOPHL) distribution, obtained by applying a Marshall–Olkin transform to the Power Half-Logistic baseline. We derive core properties—including moments, hazard rate characterization, and Rényi entropy—and develop inference under generalized progressive hybrid censoring. Estimation is carried out via maximum likelihood and Bayesian methods using a Metropolis–Hastings sampler. Asymptotic results, Fisher information, and corresponding confidence/credible intervals are provided. A Monte Carlo study assesses bias, the mean squared error, and coverage across censoring scenarios and hazard regimes. In a case study on hydroxylated fullerene degradation, MOPHL outperforms nine competing models in goodness-of-fit and predictive reliability. We also report the mean time to failure and mean residual life to support engineering decision-making. The proposed framework offers a tractable and robust tool for degradation analysis under censored data, with applicability to materials, mechanical components, biomedical devices, and environmental monitoring.

Practical Bayes filters often assume the state distribution of each time step to be Gaussian for computational tractability, resulting in the so-called Gaussian filters. When facing nonlinear systems, Gaussian filters such as extended Kalman filter (EKF) or unscented Kalman filter (UKF) typically rely on certain linearization techniques, which can introduce large estimation errors. To address this issue, this paper reconstructs the prediction and update steps of Gaussian filtering as solutions to two distinct optimization problems, whose optimal conditions are found to have analytical forms from Stein's lemma. It is observed that the stationary point for the prediction step requires calculating the first two moments of the prior distribution, which is equivalent to that step in existing moment-matching filters. In the update step, instead of linearizing the model to approximate the stationary points, we propose an iterative approach to directly minimize the update step's objective to avoid linearization errors. For the purpose of performing the steepest descent on the Gaussian manifold, we derive its natural gradient that leverages Fisher information matrix to adjust the gradient direction, accounting for the curvature of the parameter space. Combining this update step with moment matching in the prediction step, we introduce a new iterative filter for nonlinear systems called Natural Gradient Gaussian Approximation filter, or NANOfilter for short. We prove that NANO filter locally converges to the optimal Gaussian approximation at each time step. Furthermore, the estimation error is proven exponentially bounded for nearly linear measurement equation and low noise levels through constructing a supermartingale-like property across consecutive time steps. Real-world experiments demonstrate that, compared to popular Gaussian filters such as EKF, UKF, iterated EKF, and posterior linearization filter, NANO filter reduces the average root mean square error by approximately 45% while maintaining a comparable computational burden.

Abstract The SU(1,1) interferometer, a nonlinear analog of the traditional Mach-Zehnder interferometer, has emerged as a powerful tool for achieving phase sensitivity beyond the standard quantum limit. In this work, we propose the use of a superposition of even and odd coherent states as input state to enhance the phase sensitivity of an SU(1,1) interferometer. These non-classical states exhibit unique properties such as squeezing, entanglement, and quantum interference, which can be harnessed to improve metrological precision. Phase sensitivity is analyzed using single-intensity detection and homodyne detection schemes under both ideal and photon loss cases, which demonstrates significant improvements over classical and squeezed-vacuum inputs. In addition, we evaluate the quantum Cramér-Rao lower bound by employing the quantum Fisher information formalism, showing that it surpasses the standard quantum limit and approaches the Heisenberg limit under optimal conditions. Our results highlight the potential of the even and odd coherent states superposition in quantum metrology and provide a pathway for achieving ultra-precise phase measurements in SU(1,1) interferometers for applications in optical sensing, and quantum information processing.

In GNSS-denied large-scale outdoor environments, UAVs and UGVs that rely solely on visual–inertial odometry (VIO) suffer from accumulated global drift as the trajectory grows. Meanwhile, inter-platform ultra-wideband (UWB) ranging exhibits unknown, time-varying noise under NLOS/multipath, rendering naïve weighting unreliable. This paper presents an anchor-free collaborative localization framework for UAV–UGV teams that fuses pairwise UWB ranges (including UAV–UAV, UAV–UGV, and UGV–UGV) with onboard VIO in a factor-graph backend via a two-stage robust scheme. First, we bound VIO drift using per-agent state covariance and reject UWB outliers with a Mahalanobis gate, preventing early-stage bias when VIO is still accurate. Then, during global optimization, we adaptively estimate the Fisher information of UWB factors from measurement–state residuals, enabling online self-tuning of measurement confidence under time-varying SNR. Real-world experiments with three UAVs and two UGVs over multi-level rooftops and forest–open areas (~1.6 km2) show that, compared to an outlier-only variant, the proposed method further reduces localization RMSE by about 24.6% and maximum error by about 31.2% for both UAVs and UGVs, maintaining strong performance during long trajectories dominated by VIO drift and NLOS ranges. The approach requires no fixed anchors or GNSS and is applicable to UAV–UGV teams for disaster response, cooperative mapping/inspection, and bandwidth-limited operations.

Recent theoretical and experimental work has shown that the quantum Fisher information associated with estimating the separation between two optical point sources remains finite at small separations, effectively opening up new routes to super-resolution imaging of simultaneously emitting sources. Most studies to date, however, implicitly invoke the scalar approximation, which is not appropriate in the context of high-numerical-aperture microscopy. Utilizing parameter estimation theory, here we consider the estimation of separation between two closely spaced dipole emitters, a commonly employed model for single-molecule optical beacons. We consider two limiting cases: one in which the orientations of the emitters are fixed and equal, and another in which both dipoles freely sample all of orientation space over the course of the measurement. We quantify precision limits using quantum and classical variants of the Fisher information and Cramér-Rao bound. In all cases, the vectorial nature of the emission complicates the analyses, but with appropriate filtering of the collected light in the azimuthal-radial polarization basis, a previously proposed scheme to saturate the quantum Fisher information via image inversion interferometry can be salvaged.

Abstract The precision of quantum measurements can be effectively improved by using both photon-added non-Gaussian operations and Kerr nonlinear phase shift. Here, we employ a coherent state mixed with a photon-added squeezed vacuum state as the input to a Mach–Zehnder interferometer with parity detection, thereby achieving a significant enhancement in phase measurement precision. Our research focuses on the phase sensitivity of linear phase shift under both ideal conditions and photon loss, as well as quantum Fisher information (QFI). The results demonstrate that employing the photon addition operations can markedly enhance phase sensitivity and QFI, and under optimal conditions, the measurement precision can approach the Heisenberg limit for the linear phase shift case. In addition, we delve deeper into the scenario of replacing the linear phase shift with a Kerr nonlinear one and systematically analyze the QFI under both ideal and photon loss conditions. By comparison, it is evident that employing both the photon addition operations and the Kerr nonlinear phase shift can further significantly enhance phase measurement precision while effectively improving the system’s robustness against photon loss. These findings are instrumental in facilitating the development and practical application of quantum metrology.

Aided by quantum sources, quantum metrology helps enhance measurement precision. Here, we construct a theoretical model for quantum imaging based on squeezed states and present the corresponding numerical results. Through discretization and quantum Fisher information theory, we investigate the two-point resolution and spatial multi-parameter estimation of optical fields with unknown spatial distributions. We calculate and compare imaging results based on squeezed vacuum states, coherent states, and squeezed coherent states; our results show that squeezed coherent states yield greater quantum Fisher information, which can effectively improve imaging quality. In addition, we analyze the influence of imaging basis functions, degree of squeezing, quantum correlations, and other factors on imaging performance. The proposed quantum imaging model and computational method can be extended to more complex scenarios, such as multi-mode squeezed-state imaging schemes and incoherent imaging systems. In the future, this approach is expected to find applications in practical imaging systems, including Raman microscopy and stimulated Brillouin scattering imaging.

Abstract We investigate quantum sensing for spectroscopy in a system consisting of a two-level atom coupled to a continuum of modes. We focus on optimizing the pulse shape of a coherent state to maximize the quantum Fisher information (QFI) of the emitted light with the aim of estimating the atom’s dipole moment, which is proportional to its spontaneous emission rate. To achieve this, we derive a set of coupled differential equations, which include the standard optical Bloch equations as a subset and whose solution directly yields the QFI of the emitted light without resorting to finite-difference methods. Furthermore, we analyze the factors that govern its optimization, provide analytic solutions in both the long and the short pulse width limits, and examine the role of the average photon number of the pulses. We then show that under the closed (periodic) boundary conditions, the harmonic (plane-wave) with frequency equal to half the spontaneous emission rate and a phase determined by detuning are optimal in the long pulse width limit. We further show numerically that photodetection saturates the classical Fisher information.

Robust multi-innovation full parameter identification for separable fractional-order systems based on online measurements.

Abstract Quantifying measurement precision in quantum systems is vital for advancing quantum technologies such as sensing, communication, and computation. The quantum Fisher information (QFI) sets the ultimate precision bound in Hermitian systems; however, extending this concept to non-Hermitian systems, even those with real spectra, poses conceptual challenges due to their non-unitary dynamics. We compare three probability-conserving approaches for evaluating QFI in such systems: (i) simple normalization, (ii) metric formalism, and (iii) master-equation framework. Although all three ensure probability conservation, they differ in physical interpretation and in how they quantify estimation precision. Our study is particularly motivated by previous studies that have shown that the simple normalization method for non-Hermitian Hamiltonian generated dynamics may lead to misleading or even unphysical conclusions for certain quantum information theoretic tasks. We emphasize, in this article, that the metric formalism naturally enables the use of standard Hermitian metrology tools in cases where it provides a coherent and physically consistent framework for non-Hermitian systems.

Abstract Quantum geometry is a differential geometry based on quantum mechanics. It
is related to various transport and optical properties in condensed matter
physics. The Zeeman quantum geometry is a generalization of quantum geometry
including the spin degrees of freedom. It is related to electromagnetic
cross responses. Quantum geometry is generalized to non-Hermitian systems
and density matrices. Especially, the latter is quantum information
geometry, where the quantum Fisher information naturally arises as quantum
metric. We apply these results to the $X$-wave magnets, which include $d$%
-wave, $g$-wave and $i$-wave altermagnets as well as $p$-wave and $f$-wave
magnets. They have universal physics for anomalous Hall conductivity,
tunneling magneto-resistance and planar Hall effect. We also study
magneto-optical conductivity, magnetic circular dichroism and Friedel
oscillations in the $X$-wave magnets. Various analytic formulas are derived
in the case of two-band Hamiltonians. This paper presents a review of recent
progress together with some
original results

Visible light positioning (VLP) has emerged as a promising solution to address the requirements of indoor industrial localization services, such as the smart healthcare and indoor navigation. Increasing LEDs can boost positioning performance while leading to higher energy consumption, increased costs, and potential signal interference issues. To solve this problem, we propose an LED deployment algorithm that improves VLP performance by optimizing the placement of LEDs, thereby avoiding the introduction of excessive LEDs. The proposed algorithm uses the squared position error bound (SPEB) as the metric for deployment to assess the overall positioning performance. Additionally, we utilize Fisher information (FI) to quantify the information received from different LEDs and derive the fusion rules of information ellipses (IEs) to guide the deployment, aiming to maximize the overall received information in the system. To address the non-convex deployment problem, we introduce the confidence region for convex relaxation of the localization area, achieving deployment by minimizing the upper bound of SPEB within the confidence region. Moreover, we derive the localization error bounds to analyze the impact of various key parameters on positioning performance. Convincing simulation results demonstrate the significant improvement in VLP performance achieved with the proposed algorithm.

What is the cosmological information content of a cubic Gigaparsec of dark matter?Extracting cosmological information from the non-linear matter distribution has high potential to tighten parameter constraints in the era of next-generation surveys such as Euclid, DESI, and the Vera Rubin Observatory. Traditional approaches relying on summary statistics like the power spectrum and bispectrum, though analytically tractable, fail to capture the full non-Gaussian and non-linear structure of the density field. Simulation-Based Inference (SBI) provides a powerful alternative by learning directly from forward-modeled simulations. In this work, we apply SBI to the Quijote dark matter simulations and introduce a hierarchical method that integrates small-scale information from field sub-volumes or patches with large-scale statistics such as power spectrum and bispectrum. This hybrid strategy is efficient both computationally and in terms of the amount of training data required. It overcomes the memory limitations associated with full-field training. We show that our approach enhances Fisher information relative to analytical summaries and matches that of a very different approach (wavelet-based statistics), hinting that we are estimating most of the information content of the dark matter density field at the resolution of ∼ 7.8 Mpc/h.

Current methodological research on randomized controlled trial design has predominantly focused on studies with a single primary endpoint. However, many trials in practice involve multiple competing target events. The optimal designs for survival trials with competing target events have not been systematically addressed in the statistical literature. This paper fills this significant gap by developing design methodologies for randomized discrete-time-to-event trials with competing endpoints. We derive the Fisher information matrix for the general discrete-time survival model (DTSM) by transforming the original discrete-time survival data into proper multinomial responses. By introducing a cost-based generalized [Formula: see text]-optimal design criterion, we identify various types of optimal designs for estimating the treatment effects. Under the assumption of a parametric competing risks model for the underlying survival process, we demonstrate that the optimal treatment allocation scheme is critically influenced by the parameter values within this model. Our methodology is applied to the redesign of the SANAD trial, which examines withdrawal times from anti-epileptic drugs, thereby highlighting the advantages of our optimal design strategies. A key finding is that assigning subjects equally to the different groups in a two-arm DTSM trial with competing risks is generally a favorable choice, unless the hazard rates over the duration of the trial in both groups are low.

This paper provides a theoretical analysis of a Mach–Zehnder interferometer (MZI) enhanced by optical parametric amplifiers (OPA) placed inside its arms and fed by a coherent input source. Contrary to most cases found in literature, we perform a wider analysis by considering the MZI unbalanced and two internal phase shifts. We discuss the theoretical best-case phase sensitivity of this setup via the quantum Cramér–Rao bound (QCRB) and we consider scenarios having – or not – access to an external phase reference. We also assess the realistic performance of the OPA-enhanced MZI by employing a balanced homodyne detection (asymmetric and symmetric phase shifts) scheme and a difference-intensity detection scheme. We are thus able to find the optimum beam splitter (BS) transmissivity (transmission coefficient) of the first BS via the quantum Fisher information (QFI), while for the second one, we need to take the detection scheme into account. We are able to find these optimal values for all considered scenarios and thus assess the best-case phase sensitivity provided by this setup.

In complex maritime environments and scenarios with severe signal interference, unmanned surface vehicle (USV) swarms face dual challenges: unreliable GNSS signals due to interference and difficulties in accurately calibrating multi-sensor installation errors. These issues severely constrain the capability for high-precision cooperative formation operations. To address these problems, this paper proposes a cooperative localization and all-source online calibration algorithm based on a unified factor graph optimization framework. First, a tightly coupled all-source graph framework is established, integrating navigation radar, electro-optical systems (EOSs) with laser rangefinders, IMU, and GNSS into a sliding window. By leveraging high-precision mutual observations among the swarm, strong geometric constraints are constructed to mitigate the drift of individual inertial navigation systems. Second, an adaptive GNSS weighting mechanism based on signal quality and a degradation detection strategy based on eigenvalue analysis of the Fisher Information Matrix (FIM) are designed. These mechanisms enable online identification and robust estimation of extrinsic parameters, effectively resolving calibration divergence under weak excitation conditions such as straight-line sailing. Finally, the proposed algorithm is validated using field data from three USVs combined with simulated interference experiments. Results demonstrate that the algorithm can rapidly converge to high-precision calibration parameters without artificial targets (radar translation error < 0.2 m, EOS rotation error < 0.05°). During periods of simulated GNSS interference, the cooperative localization root mean square error (RMSE) is reduced to 2.85 m, representing an accuracy improvement of approximately 84.5% compared to traditional methods. This study achieves a “more accurate as it runs” cooperative navigation effect, providing reliable technical support for USV swarm applications in GNSS-denied environments.

Using an aqueous solution of acetic acid an example, this study has investigated a mathematical model of the electrical conductivity of weak electrolyte dilute solutions. To resolve the issue of identification and reliability in determining the parameters for the mathematical model of the electrical conductivity of dilute solutions of weak electrolytes, the determinant of the Fisher information matrix was calculated. The issues related to determining the association constants and the limiting molar electrical conductivity of weak electrolytes under different conditions of experimental experiments were identified and explained. This paper reports results of mathematical processing of conductometric data for aqueous solutions of acetic acid. It was established that for weak, associated electrolytes, when determining the association constants and the limiting molar coefficients, it is necessary to take into account the existence of a correlation between them. It is proven that at large values of the association constant (5.58 * 104 mol/L) the determinant of the Fisher information matrix is close to zero and there is a structural non-identification of parameters for the mathematical model of electrical conductivity of dilute solutions of weak electrolytes. It is shown that the results of mathematical processing of conductometric data for aqueous solutions of acetic acid indicate the presence of structural non-identification. This is confirmed by the values of the determinant of the Fisher information matrix, which is equal to 5.5 * 10-8, and the normalized index of 0.988. Analysis of the shape of the surface of the objective functions of the studied mathematical models and the form of the average error ellipse reveals the existence of a canyon with an almost flat bottom, which complicates the interpretation and reliability of parameters for the mathematical model of electrical conductivity. The results confirm the possibility of structural non-identification of parameters for the mathematical model of the electrical conductivity of dilute solutions of weak electrolytes

Non-Hermitian systems have attracted considerable interest over the last few decades due to their unique spectral and dynamical properties not encountered in Hermitian counterparts. An intensely debated question is whether non-Hermitian systems, described by pseudo-Hermitian Hamiltonians with real spectra, can offer enhanced sensitivity for parameter estimation when they are operated at or close to exceptional points. However, much of the current analysis and conclusions are based on mathematical formalism developed for Hermitian quantum systems, which is questionable when applied to pseudo-Hermitian Hamiltonians, whose Hilbert space is intrinsically nonflat. Here, we develop a covariant formulation of quantum Fisher information (QFI) defined on the deformed Hilbert space of pseudo-Hermitian Hamiltonians. This covariant framework ensures the preservation of the state norm and enables a consistent treatment of parameter sensitivity. We further show that the covariant QFI of pseudo-Hermitian systems is dual to the ordinary QFI of corresponding Hermitian systems. Importantly, this correspondence naturally imposes an upper bound on the covariant QFI and allows one to identify optimal projections which saturate the corresponding classical Fisher information to this ultimate limit. The developed framework also enables to set the criteria under which pseudo-Hermitian sensors can exhibit an advantage over their Hermitian counterparts of the same dimensionality.

Truss structures are widely used in actual bridge systems. During a long period of service life, the aging process of the bridges and the influence of the environment cause the measurements to be susceptible to noise during observations, thus reducing the ability of the data to effectively limit the posterior distribution of the model parameters. To overcome these issues, a numerical model based on OpenSees is adopted as the forward model to identify the parameter-response sensitivity matrix using the finite difference method. Under the assumption that the measurements follow a Gaussian distribution, a Fisher information matrix is introduced to characterize the information content of different candidate schemes for ranking and selecting loading and measurement schemes. Then, the Bayesian parameter updates based on the selected representative schemes are performed, while posterior degeneracy for the parameters is calculated using Monte Carlo sampling methods. The efficacy of the selected schemes is demonstrated by comparing the ability to reduce uncertainties and parameter correlations.