Finite-dimensional Matrix Research Articles

An algebraic formulation of nonassociative quantum mechanics

We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop Gel’fand–Naimark–Segal constructions. We describe Heisenberg and Schrödinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra.

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

Computation of prolate spheroidal wavefunctions (PSWFs) is notoriously difficult and time consuming. This paper applies operator theory to the discrete Fourier transform (DFT) to address the problem of computing PSWFs. The problem is turned into an infinite dimensional matrix operator eigenvalue problem, which we recognize as being the definition of the DPSSs. Truncation of the infinite matrix leads to a finite dimensional matrix eigenvalue problem which in turn yields what is known as the Slepian basis. These discrete-valued Slepian basis vectors can then be used as (approximately) discrete time evaluations of the PSWFs. Taking an inverse Fourier transform further demonstrates that continuous PSWFs can be reconstructed from the Slepian basis. The feasibility of this approach is shown via theoretical derivations followed by simulations to consider practical aspects. Simulations demonstrate that the level of errors between the reconstructed Slepian basis approach and true PSWFs are low when the orders of the eigenvectors are low but can become large when the orders of the eigenvectors are high. Accuracy can be increased by increasing the number of points used to generate the Slepian basis. Users need to balance accuracy with computational cost. For large time-bandwidth product PSWFs, the number of Slepian basis points required increases for a reconstruction to reach the same error as for low time-bandwidth products. However, when the time-bandwidth products increase and reach maximum concentration, the required number of points to achieve a given error level achieves steady state values. Furthermore, this method of reconstructing the PSWF from the Slepian basis can be more accurate when compared to the Shannon sampling approach and traditional quadrature approach for large time-bandwidth products. Finally, since the Slepian basis represents the (approximate) sampled values of PSWFs, when the number of points is sufficiently large, the reconstruction process can be omitted entirely so that the Slepian vectors can be used directly, without a reconstruction step.

A note on factorization of vector valued functions involving the extraction of elementary factors

This note contains reflections on a type of factorization that came up in the work of analytic operator valued functions by I.C. Gohberg and E.I. Sigal (cf. [14], [10], [11]). The determining characteristic of the factorizations in question is that they are obtained by the extraction of elementary factors of the form I−P+λP where P is an idempotent in the underlying Banach algebra. It will be demonstrated here that there is a fundamental difference between the finite dimensional (matrix) case and the situation where infinite dimensional Banach algebras are taken into consideration.

Zonotopic set-membership estimation for time-varying systems subject to dynamical biases and quantization effects

In this paper, the zonotopic set-membership estimation (SME) problem is studied for time-varying systems subject to dynamical biases and uniform quantization. The considered dynamical bias evolves according to a certain dynamic process disturbed by unknown-but-bounded noises. The main objective of the paper is to propose a zonotopic SME method such that the real states are included in zonotopes described by linear image of an unit hypercube. By analyzing the dynamics of biases and states, the intersection set containing the state are first obtained based on the quantization measurements. Then, an auxiliary zonotope is constructed to bound the intersection where the center and generator matrix are presented recursively. Through orchestrating the correlation matrix, the F-Radius of such an auxiliary zonotope is further minimized. To reduce the computational burden, the desired zonotope with finite-dimensional generator matrix is constructed to externally approximate the auxiliary zonotope. Furthermore, the boundedness stability is analyzed for the developed zonotopic SME approach in the simulations presence of dynamical biases and uniform quantization. Finally, an example is provided to display the effectiveness of the SME method.

3 definitions of BF theory on homology 3-spheres

3-dimensional BF theory with gauge group G (= Chern-Simons theory with non-compact gauge group TG) is a deceptively simple yet subtle topological gauge theory. Formally, its partition function is a sum/integral over the moduli space mathcal{M} of flat connections, weighted by the Ray-Singer torsion. In practice, however, this formal expression is almost invariably singular and ill-defined.In order to improve upon this, we perform a direct evaluation of the path integral for certain classes of 3-manifolds (namely integral and rational Seifert homology spheres). By a suitable choice of gauge, we sidestep the issue of having to integrate over mathcal{M} and reduce the partition function to a finite-dimensional Abelian matrix integral which, however, itself requires a definition. We offer 3 definitions of this integral, firstly via residues, and then via a large k limit of the corresponding G × G or Gℂ Chern-Simons matrix integrals (obtained previously). We then check and discuss to which extent the results capture the expected sum/integral over all flat connections.

Data assimilation in operator algebras

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture

I show that if a finite-dimensional density matrix has strictly smaller von Neumann entropy than a second one of the same dimension (and the rank is not bigger), then sufficiently (but finitely) many tensor-copies of the first density matrix majorize a density matrix whose single-body marginals are all exactly equal to the second density matrix. This implies an affirmative solution of the exact catalytic entropy conjecture (CEC) introduced by Boes et al. [PRL 122, 210402 (2019)]. Both the Lemma and the solution to the CEC transfer to the classical setting of finite-dimensional probability vectors (with permutations of entries instead of unitary transformations for the CEC).

Complete Entropic Inequalities for Quantum Markov Chains

We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or L_2-estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.

On orthogonal bases in the Hilbert-Schmidt space of matrices

Decomposition of (finite-dimensional) operators in terms of orthogonal bases of matrices has been a standard method in quantum physics for decades. In recent years, it has become increasingly popular because of various methodologies applied in quantum information, such as the graph state formalism and the theory of quantum error correcting codes, but also due to the intensified research on the Bloch representation of quantum states. In this contribution we collect various interesting facts and identities that hold for finite-dimensional orthogonal matrix bases.

A pseudospectral method for investigating the stability of linear population models with two physiological structures.

The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. In this paper, we propose a general numerical method to approximate this spectrum. In particular, we first reformulate the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.

Bivariate collocation for computing R0 in epidemic models with two structures

Structured epidemic models can be formulated as first-order hyperbolic PDEs, where the “spatial” variables represent individual traits, called structures. For models with two structures, we propose a numerical technique to approximate R0, which measures the transmissibility of an infectious disease and, rigorously, is defined as the dominant eigenvalue of a next generation operator. Via bivariate collocation and cubature on tensor grids, the latter is approximated with a finite-dimensional matrix, so that its dominant eigenvalue can easily be computed with standard techniques. We use test examples to investigate experimentally the behavior of the approximation: the convergence order appears to be infinite when the corresponding eigenfunction is smooth, and finite for less regular eigenfunctions. To demonstrate the effectiveness of the technique for more realistic applications, we present a new epidemic model structured by demographic age and immunity, and study the approximation of R0 in some particular cases of interest.

The Laplace transform of the integrated Volterra Wishart process

AbstractWe establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore, we link the characteristic exponents to a system of non‐standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear‐quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.

Deformed contour segment matching for multi-source images

Robust and accurate multi-source matching is a difficult task due to significant nonlinear radiometric differences, background clutter, and geometric deformation in corresponding regions. Motivated by these existing problems, a discriminating yet robust combined descriptor for multi-source image matching, called deformed contour segment similarity (DCSS), is proposed in this work. First, the proposed DCSS, which is constructed by histogram of the combined contour features rather than the commonly used corner point and gradient, presents the accurate correspondence between image pairs and improves the descriptive ability to radiometric differences. Second, the deformed curve is presented via a finite-dimensional matrix Lie group to determine the similarity metric with an explicit geodesic solution. The geodesic distance, which indicates the nearest distance between curves in fluid space, is defined as the weight coefficient of the constructed histogram to enhance the robustness of the descriptor. The proposed algorithm utilizes the holistic contour information for the scoring and ranking of the shape similarity hypothesis, which can effectively reduce the influence of partially missing contours. Finally, a precise bilateral matching rule is used to perform the matching between the corresponding contour segments. Some experiments are carried out on various infrared-visible image data sets. The results demonstrate that the proposed DCSS achieves more robust and accurate matching performance than many popular multi-source image matching methods.

On the decoupled Markov group conjecture

The Markov group conjecture, a long-standing open problem in the theory of Markov processes with countable state space, asserts that a strongly continuous Markov semigroup $T = (T_t)_{t \in [0,\infty)}$ on $\ell^1$ has bounded generator if the operator $T_1$ is bijective. Attempts to disprove the conjecture have often aimed at glueing together finite dimensional matrix semigroups of growing dimension - i.e., it was tried to show that the Markov group conjecture is false even for Markov processes that decouple into (infinitely many) finite dimensional systems. In this article we show that such attempts must necessarily fail, i.e., we prove the Markov group conjecture for processes that decouple in the way described above. In fact, we even show a more general result that gives a universal norm estimate for bounded generators $Q$ of positive semigroups on any Banach lattice. Our proof is based on a filter product technique, infinite dimensional Perron-Frobenius theory and Gelfand's $T = \operatorname{id}$ theorem.

Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions

We prove a version of Talagrand’s concentration inequality for subordinated sub-Laplacians on a compact Riemannian manifold using tools from noncommutative geometry. As an application, motivated by quantum information theory, we show that on a finite-dimensional matrix algebra the set of self-adjoint generators satisfying a tensor stable modified logarithmic Sobolev inequality is dense.

Globally Exponentially Convergent Continuous Observers for Velocity Bias and State for Invariant Kinematic Systems on Matrix Lie Groups

In this article globally exponentially convergent continuous observers for invariant kinematic systems on finite-dimensional matrix Lie groups has been proposed. Such an observer estimates, from measurements of landmarks, vectors, and biased velocity, both the system state and the unknown constant bias in velocity measurement, where the state belongs to the state-space Lie group and the velocity to the Lie algebra of the Lie group. The main technique is to embed a given system defined on a matrix Lie group into Euclidean space and build observers in the Euclidean space. The theory is illustrated with the special Euclidean group in three dimensions, and it is shown that the observer works well even in the presence of measurement noise.

Characterization of Equivariant Maps and Application to Entanglement Detection

We study equivariant linear maps between finite-dimensional matrix algebras, as introduced by Bhat. These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitary conjugation as studied by Bhat and as a generalization of the equivariant maps studied by Collins et al. Using representation theory, we fully compute them and study their graphical representation, and show that they are basically enough to study all equivariant maps. We finally apply them to the problem of entanglement detection and prove that they form a sufficient (infinite) family of positive maps to detect all k-entangled density matrices.

A Nonlinear Convergence Consensus: Extreme Doubly Stochastic Quadratic Operators for Multi-Agent Systems

We investigate a novel nonlinear consensus from the extreme points of doubly stochastic quadratic operators (EDSQO), based on majorization theory and Markov chains for time-varying multi-agent distributed systems. We describe a dynamic system that has a local interaction network among agents. EDSQO has been applied for distributed agent systems, on a finite dimensional stochastic matrix. We prove that multi-agent systems converge at a center (common value) via the extreme waited value of doubly stochastic quadratic operators (DSQO), which are only 1 or 0 or 1/2 1 2 if the exchanges of each agent member has no selfish communication. Applying this rule means that the consensus is nonlinear and low-complexity computational for fast time convergence. The investigated nonlinear model of EDSQO follows the structure of the DeGroot linear (DGL) consensus model. However, EDSQO is nonlinear and faster convergent than the DGL model and is of lower complexity than DSQO and cubic stochastic quadratic operators (CSQO). The simulation result and theoretical proof are illustrated.

Primary decompositions of unital locally matrix algebras

We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser. 22(64)(3) (1948) 443–454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. We also show that for an arbitrary infinite Steinitz number [Formula: see text] there exists a unital locally matrix algebra [Formula: see text] having the Steinitz number [Formula: see text] and not isomorphic to a tensor product of finite-dimensional matrix algebras.

Solvability, Completeness and Computational Analysis of A Perturbed Control Problem with Delays

As a first step, we provide a precise mathematical framework for the class of control problems with delays (which we refer to as the control problem) under investigation in a Banach space setting, followed by careful definitions of the key properties to be analyzed such as solvability and complete controllability. Then, we recast the control problem in a reduced form that is especially amenable to the innovative analytical approach that we employ. We then study in depth the solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters. The main tool in our approach is the use of a Borsuk–Ulam type fixed point theorem to analyze the topological structure of a suitably reduced control problem solution, with a focus on estimating the dimension of the corresponding solution set, and proving its completeness. Next, we investigate its analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. Then, we describe a novel computational projection-based discretization scheme of our own devising for obtaining accurate approximate solutions of the control problem along with useful error estimates. The scheme effectively reduces the infinite-dimensional problem to a sequence of solvable finite-dimensional matrix valued tasks. Finally, we include an application of the scheme to a special degenerate case of the problem wherein the Banach–Steinhaus theorem is brought to bear in the estimation process.