Fisher information

We study self-organization of collective motion as a thermodynamic phenomenon in the context of the first law of thermodynamics. It is expected that the coherent ordered motion typically self-organises in the presence of changes in the (generalized) internal energy and of (generalized) work done on, or extracted from, the system. We aim to explicitly quantify changes in these two quantities in a system of simulated self-propelled particles and contrast them with changes in the system's configuration entropy. In doing so, we adapt a thermodynamic formulation of the curvatures of the internal energy and the work, with respect to two parameters that control the particles' alignment. This allows us to systematically investigate the behavior of the system by varying the two control parameters to drive the system across a kinetic phase transition. Our results identify critical regimes and show that during the phase transition, where the configuration entropy of the system decreases, the rates of change of the work and of the internal energy also decrease, while their curvatures diverge. Importantly, the reduction of entropy achieved through expenditure of work is shown to peak at criticality. We relate this both to a thermodynamic efficiency and the significance of the increased order with respect to a computational path. Additionally, this study provides an information-geometric interpretation of the curvature of the internal energy as the difference between two curvatures: the curvature of the free entropy, captured by the Fisher information, and the curvature of the configuration entropy.

We prove the global existence of non-negative variational solutions to the “drift diffusion” evolution equation \({{\partial_t} u+ div \left(u{\mathrm{D}}\left(2 \frac{\Delta \sqrt u}{\sqrt u}-{f}\right)\right)=0}\) under variational boundary condition. Despite the lack of a maximum principle for fourth order equations, non-negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the (perturbed) Fisher information functional \({\fancyscript F^f(u):=\frac 12\int \left|{\mathrm{D}} \log u\right|^2 {u} dx+\int fu dx}\) with respect to the Kantorovich–Rubinstein–Wasserstein distance between probability measures. We also study long-time behavior of the solutions, proving their exponential decay to the equilibrium state g = e−V characterized by \({-\Delta V+\frac12 \left|{\mathrm{D}} V\right|^2=f,\quad \int {\rm e}^{-V} dx=\int u_{0}dx,}\) when the potential V is uniformly convex: in this case the functional \({\fancyscript F^f}\) coincides with the relative Fisher information \({\fancyscript F^f(u)=\frac12\fancyscript I(u|g)= \int \left|{\mathrm{D}}\log(u/g)\right|^2u dx}\) .

This paper focuses on the stochastic Cramer-Rao bound (CRB) for the direction of arrival (DOA) for binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulated signals observed on the background of nonuniform white noise with an arbitrary diagonal covariance matrix. Explicit expressions of the CRB for the DOA parameter alone in the case of a single signal waveform is

It is known that equilibrium thermodynamics can be deduced from a constrained Fisher information extemizing process. We show here that, more generally, both nonequilibrium and equilibrium thermodynamics can be obtained from such a Fisher treatment. Equilibrium thermodynamics corresponds to the ground-state solution, and nonequilibrium thermodynamics corresponds to excited-state solutions, of a Schrödinger wave equation (SWE). That equation appears as an output of the constrained variational process that extremizes Fisher information. Both equilibrium and nonequilibrium situations can thereby be tackled by one formalism that clearly exhibits the fact that thermodynamics and quantum mechanics can both be expressed in terms of a formal SWE, out of a common informational basis. As an application, we discuss viscosity in dilute gases.

The Fisher information $F$ gives a limit to the ultimate precision achievable in a phase estimation protocol. It has been shown recently that the Fisher information for a linear two-mode interferometer cannot exceed the number of particles if the input state is separable. As a direct consequence, with such input states the shot-noise limit is the ultimate limit of precision. In this work, we go a step further by deducing bounds on $F$ for several multiparticle entanglement classes. These bounds imply that genuine multiparticle entanglement is needed for reaching the highest sensitivities in quantum interferometry. We further compute similar bounds on the average Fisher information $\bar F$ for collective spin operators, where the average is performed over all possible spin directions. We show that these criteria detect different sets of states and illustrate their strengths by considering several examples, also using experimental data. In particular, the criterion based on $\bar F$ is able to detect certain bound entangled states.

The theory of Extreme Physical Information (EPI) is used to deduce a probability density function (PDF) of a system that exhibits a power law tail. The computed PDF is useful to study and fit several observed distributions in complex systems. With this new approach it is possible to describe extreme and rare events in the tail, and also the frequent events in the distribution head. Using EPI, an information functional is constructed, and minimized using Euler-Lagrange equations. As a solution, a second order differential equation is derived. By solving this equation a family of functions is calculated. Using these functions it is possible to describe the system in terms of eigenstates. A dissipative term is introduced into the model, as a relevant term for the study of open systems. One of the main results is a mathematical relation between the scaling parameter of the power law observed in the tail and the shape of the head.

This paper presents a novel application of particle swarm optimization (PSO) in combination with another computational intelligence (CI) technique, namely, proximal support vector machine (PSVM) for machinery fault detection. Both real-valued and binary PSO algorithms have been considered along with linear and nonlinear versions of PSVM. The time domain vibration signals of a rotating machine with normal and defective bearings are processed for feature extraction. The features extracted from original and preprocessed signals are used as inputs to the classifiers (PSVM) for detection of machine condition. Input features are selected using a PSO algorithm. The classifiers are trained with a subset of experimental data for known machine conditions and are tested using the remaining data. The procedure is illustrated using the experimental vibration data of a rotating machine. The influences of the number of features, PSO algorithms and type of classifiers (linear or nonlinear PSVM) on the detection success are investigated. Results are compared with a genetic algorithm (GA) and principal component analysis (PCA). The PSO based approach gave test classification success above 90% which were comparable with the GA and much better than PCA. The results show the effectiveness of the selected features and classifiers in detection of machine condition.

Abstract: We introduce an operational interpretation for pure-state global multipartite entanglement based on quantum estimation. We show that the estimation of the strength of low-noise locally depolarizing channels, as quantified by the regularized quantum Fisher information, is ...

Abstract: A quantum entropy space is suggested as the fundamental arena describing the
quantum effects. In the quantum regime the entropy is expressed as the superposition of
many different Boltzmann entropies that span the space of the entropies before any
measure. When a measure is performed the quantum entropy collapses to one component.
A suggestive reading of the relational interpretation of quantum mechanics and of Bohm’s
quantum potential in terms of the quantum entropy are provided. The space associated with
the quantum entropy determines a distortion in the classical space of position, which
appears as a Weyl-like gauge potential connected with Fisher information. This Weyl-like
gauge potential produces a deformation of the moments which changes the classical action
in such a way that Bohm’s quantum potential emerges as consequence of the non classical
definition of entropy, in a non-Euclidean information space under the constraint of a
minimum condition of Fisher information (Fisher Bohm- entropy). Finally, the possible
quantum relativistic extensions of the theory and the connections with the problem of
quantum gravity are investigated. The non classical thermodynamic approach to quantum
phenomena changes the geometry of the particle phase space. In the light of the
representation of gravity in ordinary phase space by torsion in the flat space (Teleparallel
gravity), the change of geometry in the phase space introduces quantum phenomena in a
natural way. This gives a new force to F. Shojai’s and A. Shojai’s theory where the
geometry of space-time is highly coupled with a quantum potential whose origin is not the
Schrödinger equation but the non classical entropy of a system of many particles that together change the geometry of the phase space of the positions (entanglement). In this
way the non classical thermodynamic changes the classical geodetic as a consequence of
the quantum phenomena and quantum and gravity are unified. Quantum affects geometry
of multidimensional phase space and gravity changes in any point the torsion in the
ordinary four-dimensional Lorenz space-time metric

Rather than make a classification decision (pass/fail, below basic/basic/proficient/advanced) for an individual after administering a fixed number of items, it is possible to sequentially select items to maximize information, update the estimated classification probabilities and then evaluate whether there is enough information to terminate testing. In measurement this is frequently called adaptive or tailored testing. In statistics, this is called sequential testing.

We discuss different statistical distances in probability space, with emphasis on the Jensen-Shannon divergence, vis-a-vis {\it metrics} in Hilbert space and their relationship with Fisher's information measure. This study provides further reconfirmation of Wootters' hypothesis concerning the possibility that statistical fluctuations in the outcomes of measurements be regarded (at least partly) as responsible for the Hilbert-space structure of quantum mechanics.