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17 pages, 321 KiB

Open AccessArticle

Symmetries of Multipartite Weyl Quantum Channels

by Dariusz Chruściński, Bihalan Bhattacharya and Saikat Patra

Symmetry 2025, 17(6), 943; https://doi.org/10.3390/sym17060943 - 13 Jun 2025

Viewed by 337

Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated [...] Read more.

Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated qubit Pauli channels. Moreover, they are covariant with respective to the finite group generated by Weyl operators. In what follows, we study self-adjoint Weyl channels by providing a special Hermitian representation. For a prime dimension of the corresponding Hilbert space, the self-adjoint Weyl channels contain well-known generalized Pauli channels as a special case. We propose multipartite generalization of Weyl channels. In particular, we analyze the power of prime dimensions using finite fields and study the covariance properties of these objects. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Quantum Models)

30 pages, 2290 KiB

Open AccessArticle

Numerical Evidence for a Bipartite Pure State Entanglement Witness from Approximate Analytical Diagonalization

by Paul M. Alsing and Richard J. Birrittella

Foundations 2025, 5(2), 19; https://doi.org/10.3390/foundations5020019 - 4 Jun 2025

Viewed by 945

Abstract

We show numerical evidence for a bipartite

d \times d

pure state entanglement witness that is readily calculated from the wavefunction coefficients directly, without the need for the numerical computation of eigenvalues. This is accomplished by using an approximate analytic diagonalization of the [...] Read more.

We show numerical evidence for a bipartite

d \times d

pure state entanglement witness that is readily calculated from the wavefunction coefficients directly, without the need for the numerical computation of eigenvalues. This is accomplished by using an approximate analytic diagonalization of the bipartite state that captures dominant contributions to the negativity of the partially transposed state. We relate this entanglement witness to the Log Negativity, and show that it exactly agrees with it for the class of pure states whose quantum amplitudes form a positive Hermitian matrix. In this case, the Log Negativity is given by the negative logarithm of the purity of the amplitudes considered a density matrix. In other cases, the witness forms a lower bound to the exact, numerically computed Log Negativity. The formula for the approximate Log Negativity achieves equality with the exact Log Negativity for the case of an arbitrary pure state of two qubits, which we show analytically. We compare these results to a witness of entanglement given by the linear entropy. Finally, we explore an attempt to extend these pure state results to mixed states. We show that the Log Negativity for this approximate formula is exact for the class of pure state decompositions, for which the quantum amplitudes of each pure state form a positive Hermitian matrix. Full article

(This article belongs to the Section Mathematical Sciences)

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81 pages, 2075 KiB

Open AccessReview

A Comprehensive Review on Solving the System of Equations AX = C and XB = D

by Qing-Wen Wang, Zi-Han Gao and Jia-Le Gao

Symmetry 2025, 17(4), 625; https://doi.org/10.3390/sym17040625 - 21 Apr 2025

Cited by 3 | Viewed by 402

Abstract

This survey provides a review of the theoretical research on the classic system of matrix equations

A X = C

and

X B = D

, which has wide-ranging applications across fields such as control theory, optimization, image processing, and robotics. The paper [...] Read more.

This survey provides a review of the theoretical research on the classic system of matrix equations

A X = C

and

X B = D

, which has wide-ranging applications across fields such as control theory, optimization, image processing, and robotics. The paper discusses various solution methods for the system, focusing on specialized approaches, including generalized inverse methods, matrix decomposition techniques, and solutions in the forms of Hermitian, extreme rank, reflexive, and conjugate solutions. Additionally, specialized solving methods for specific algebraic structures, such as Hilbert spaces, Hilbert

C^{*}

-modules, and quaternions, are presented. The paper explores the existence conditions and explicit expressions for these solutions, along with examples of their application in color images. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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18 pages, 283 KiB

Open AccessArticle

A System of Coupled Matrix Equations with an Application over the Commutative Quaternion Ring

by Xiao-Quan Chen, Long-Sheng Liu, Xiao-Xiao Ma and Qian-Wen Long

Symmetry 2025, 17(4), 619; https://doi.org/10.3390/sym17040619 - 18 Apr 2025

Viewed by 263

Abstract

In this paper, we study the necessary and sufficient conditions for a system of matrix equations to have a solution and a Hermitian solution. As an application, we establish the necessary and sufficient conditions for a classical matrix system to have a reducible [...] Read more.

In this paper, we study the necessary and sufficient conditions for a system of matrix equations to have a solution and a Hermitian solution. As an application, we establish the necessary and sufficient conditions for a classical matrix system to have a reducible solution. Finally, we present an algorithm, along with two concrete examples to validate the main conclusions. Full article

(This article belongs to the Special Issue Advance in Functional Equations, Second Edition)

14 pages, 495 KiB

Open AccessArticle

A Fast Projected Gradient Algorithm for Quaternion Hermitian Eigenvalue Problems

by Shan-Qi Duan and Qing-Wen Wang

Mathematics 2025, 13(6), 994; https://doi.org/10.3390/math13060994 - 18 Mar 2025

Cited by 3 | Viewed by 464

Abstract

In this paper, based on the novel generalized Hamilton-real (GHR) calculus, we propose for the first time a quaternion Nesterov accelerated projected gradient algorithm for computing the dominant eigenvalue and eigenvector of quaternion Hermitian matrices. By introducing momentum terms and look-ahead updates, the [...] Read more.

In this paper, based on the novel generalized Hamilton-real (GHR) calculus, we propose for the first time a quaternion Nesterov accelerated projected gradient algorithm for computing the dominant eigenvalue and eigenvector of quaternion Hermitian matrices. By introducing momentum terms and look-ahead updates, the algorithm achieves a faster convergence rate. We theoretically prove the convergence of the quaternion Nesterov accelerated projected gradient algorithm. Numerical experiments show that the proposed method outperforms the quaternion projected gradient ascent method and the traditional algebraic methods in terms of computational accuracy and runtime efficiency. Full article

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27 pages, 1332 KiB

Open AccessArticle

Matrix-Sequences of Geometric Means in the Case of Hidden (Asymptotic) Structures

by Danyal Ahmad, Muhammad Faisal Khan and Stefano Serra-Capizzano

Mathematics 2025, 13(3), 393; https://doi.org/10.3390/math13030393 - 24 Jan 2025

Viewed by 870

Abstract

In the current work, we analyze the spectral distribution of the geometric mean of two or more matrix-sequences constituted by Hermitian positive definite matrices, under the assumption that all input matrix-sequences belong to the same Generalized Locally Toeplitz (GLT) ∗-algebra. We consider the [...] Read more.

In the current work, we analyze the spectral distribution of the geometric mean of two or more matrix-sequences constituted by Hermitian positive definite matrices, under the assumption that all input matrix-sequences belong to the same Generalized Locally Toeplitz (GLT) ∗-algebra. We consider the geometric mean for two matrices, using the Ando-Li-Mathias (ALM) definition, and then we pass to the extension of the idea to more than two matrices by introducing the Karcher mean. While there is no simple formula for the geometric mean of more than two matrices, iterative methods from the literature are employed to compute it. The main novelty of the work is the extension of the study in the distributional sense when input matrix-sequences belong to one of the GLT ∗-algebras. More precisely, we show that the geometric mean of more than two positive definite GLT matrix-sequences forms a new GLT matrix-sequence, with the GLT symbol given by the geometric mean of the individual symbols. Numerical experiments are reported concerning scalar and block GLT matrix-sequences in both one-dimensional and two-dimensional cases. A section with conclusions and open problems ends the current work. Full article

(This article belongs to the Special Issue Numerical Analysis and Matrix Computations: Theory and Applications)

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13 pages, 352 KiB

Open AccessArticle

A Robust Hermitian and Skew-Hermitian Based Multiplicative Splitting Iterative Method for the Continuous Sylvester Equation

by Mohammad Khorsand Zak and Abbas Abbaszadeh Shahri

Mathematics 2025, 13(2), 318; https://doi.org/10.3390/math13020318 - 20 Jan 2025

Cited by 2 | Viewed by 925

Abstract

For solving the continuous Sylvester equation, a class of Hermitian and skew-Hermitian based multiplicative splitting iteration methods is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations, and it can be equivalently written as two [...] Read more.

For solving the continuous Sylvester equation, a class of Hermitian and skew-Hermitian based multiplicative splitting iteration methods is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations, and it can be equivalently written as two multiplicative splitting matrix equations. When both coefficient matrices in the continuous Sylvester equation are (non-symmetric) positive semi-definite, and at least one of them is positive definite, we can choose Hermitian and skew-Hermitian (HS) splittings of matrices A and B in the first equation, and the splitting of the Jacobi iterations for matrices A and B in the second equation in the multiplicative splitting iteration method. Convergence conditions of this method are studied in depth, and numerical experiments show the efficiency of this method. Moreover, by numerical computation, we show that multiplicative splitting can be used as a splitting preconditioner and induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the continuous Sylvester equation. Full article

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21 pages, 662 KiB

Open AccessArticle

On Rayleigh Quotient Iteration for the Dual Quaternion Hermitian Eigenvalue Problem

by Shan-Qi Duan, Qing-Wen Wang and Xue-Feng Duan

Mathematics 2024, 12(24), 4006; https://doi.org/10.3390/math12244006 - 20 Dec 2024

Cited by 5 | Viewed by 887

Abstract

The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the use of Rayleigh quotient iteration (RQI) for solving the right eigenpairs of dual quaternion Hermitian matrices. Combined with [...] Read more.

The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the use of Rayleigh quotient iteration (RQI) for solving the right eigenpairs of dual quaternion Hermitian matrices. Combined with dual representation, the RQI algorithm can effectively compute the eigenvalue along with the associated eigenvector of the dual quaternion Hermitian matrices. Furthermore, by utilizing the minimal residual property of the Rayleigh quotient, a convergence analysis of the Rayleigh quotient iteration is derived. Numerical examples are provided to illustrate the high accuracy and low CPU time cost of the proposed Rayleigh quotient iteration compared with the power method for solving the dual quaternion Hermitian eigenvalue problem. Full article

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15 pages, 289 KiB

Open AccessArticle

Soliton Solutions to Sasa–Satsuma-Type Modified Korteweg–De Vries Equations by Binary Darboux Transformations

by Wen-Xiu Ma

Mathematics 2024, 12(23), 3643; https://doi.org/10.3390/math12233643 - 21 Nov 2024

Cited by 24 | Viewed by 986

Abstract

Sasa–Satsuma (SS)-type integrable matrix modified Korteweg–de Vries (mKdV) equations are derived from two group constraints, involving the replacement of the spectral matrix in the Ablowitz–Kaup–Newell–Segur matrix eigenproblems with its matrix transpose and its Hermitian transpose. Using the Lax pairs and dual Lax pairs [...] Read more.

Sasa–Satsuma (SS)-type integrable matrix modified Korteweg–de Vries (mKdV) equations are derived from two group constraints, involving the replacement of the spectral matrix in the Ablowitz–Kaup–Newell–Segur matrix eigenproblems with its matrix transpose and its Hermitian transpose. Using the Lax pairs and dual Lax pairs of matrix eigenproblems as a foundation, binary Darboux transformations are constructed. These transformations, initiated with a zero seed solution, facilitate the generation of soliton solutions for the SS-type integrable matrix mKdV equations presented. Full article

21 pages, 6346 KiB

Open AccessArticle

Novel Steganographic Method Based on Hermitian Positive Definite Matrix and Weighted Moore–Penrose Inverses

by Selver Pepić, Muzafer Saračević, Aybeyan Selim, Darjan Karabašević, Marija Mojsilović, Amor Hasić and Pavle Brzaković

Appl. Sci. 2024, 14(22), 10174; https://doi.org/10.3390/app142210174 - 6 Nov 2024

Cited by 1 | Viewed by 1024

Abstract

In this paper, we describe the concept of a new data-hiding technique for steganography in RGB images where a secret message is embedded in the blue layer of specific bytes. For increasing security, bytes are chosen randomly using a random square Hermitian positive [...] Read more.

In this paper, we describe the concept of a new data-hiding technique for steganography in RGB images where a secret message is embedded in the blue layer of specific bytes. For increasing security, bytes are chosen randomly using a random square Hermitian positive definite matrix, which is a stego-key. The proposed solution represents a very strong key since the number of variants of positive definite matrices of order 8 is huge. Implementing the proposed steganographic method consists of splitting a color image into its R, G, and B channels and implementing two segments, which take place in several phases. The first segment refers to embedding a secret message in the carrier (image or text) based on the unique absolute elements values of the Hermitian positive definite matrix. The second segment refers to extracting a hidden message based on a stego-key generated based on the Hermitian positive definite matrix elements. The objective of the data-hiding technique using a Hermitian positive definite matrix is to embed confidential or sensitive data within cover media (such as images, audio, or video) securely and imperceptibly; by doing so, the hidden data remain confidential and tamper-resistant while the cover media’s visual or auditory quality is maintained. Full article

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24 pages, 288 KiB

Open AccessArticle

Some New Algebraic Method Developments in the Characterization of Matrix Equalities

by Yongge Tian

Axioms 2024, 13(10), 657; https://doi.org/10.3390/axioms13100657 - 24 Sep 2024

Viewed by 907

Abstract

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and [...] Read more.

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc. Full article

(This article belongs to the Special Issue Advances in Linear Algebra with Applications)

10 pages, 6069 KiB

Open AccessArticle

Robust Sensing Based on Exceptional Points in Detuned Non-Hermitian Scattering System

by Jipeng Xu, Yuanhao Mao, Ken Liu and Zhihong Zhu

Photonics 2024, 11(9), 882; https://doi.org/10.3390/photonics11090882 - 20 Sep 2024

Cited by 1 | Viewed by 1337

Abstract

Non-Hermitian optics has revealed a series of counterintuitive phenomena with profound implications for sensing, lasing, and light manipulation. While the non-Hermiticity of Hamiltonians has been intensively investigated, recent advancements in the non-Hermitian scattering matrix have given birth to a lot of unique phenomena, [...] Read more.

Non-Hermitian optics has revealed a series of counterintuitive phenomena with profound implications for sensing, lasing, and light manipulation. While the non-Hermiticity of Hamiltonians has been intensively investigated, recent advancements in the non-Hermitian scattering matrix have given birth to a lot of unique phenomena, such as simultaneous lasing and anti-lasing, reflectionless scattering modes (RSMs), and coherent chaos control. Despite these developments, the investigation has predominantly focused on static and symmetric configurations, leaving the dynamic properties of non-Hermitian scattering in detuned systems, which is essential for applications in sensing and beyond, largely unexplored. Here, we extend the stationary behaviors associated with the RSMs to resonant detuned systems. Contrary to the common belief of exceptional point (EP) sensors as being susceptible to parametric disturbances, we induce an RSM EP in a one-dimensional optical cavity and demonstrate its robustness in displacement sensing against laser frequency drifts up to 10 MHz. Our findings not only contribute to the broader understanding of non-Hermitian scattering phenomena but also pave the way for the next generation of non-Hermitian sensors. Full article

(This article belongs to the Section Lasers, Light Sources and Sensors)

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16 pages, 279 KiB

Open AccessArticle

Solving the QLY Least Squares Problem of Dual Quaternion Matrix Equation Based on STP of Dual Quaternion Matrices

by Ruyu Tao, Ying Li, Mingcui Zhang, Xiaochen Liu and Musheng Wei

Symmetry 2024, 16(9), 1117; https://doi.org/10.3390/sym16091117 - 28 Aug 2024

Cited by 1 | Viewed by 1161

Abstract

Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation [...] Read more.

Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation

A X B + C Y D \approx E

, where X, Y are unknown dual quaternion matrices with special structures. First, we define a semi-tensor product of dual quaternion matrices and study its properties, which can be used to achieve the equivalent form of the dual quaternion matrix equation. Then, by using the dual representation of dual quaternion and the

GH

-representation of special dual quaternion matrices, we study the expression of QLY least squares Hermitian solution of the dual quaternion matrix equation

A X B + C Y D \approx E

. The algorithm is given and the numerical examples are provided to illustrate the efficiency of the method. Full article

(This article belongs to the Special Issue Exploring Symmetry in Dual Quaternion Matrices and Matrix Equations)

19 pages, 293 KiB

Open AccessArticle

Extensions of Some Statistical Concepts to the Complex Domain

by Arak M. Mathai

Axioms 2024, 13(7), 422; https://doi.org/10.3390/axioms13070422 - 22 Jun 2024

Viewed by 780

Abstract

This paper deals with the extension of principal component analysis, canonical correlation analysis, the Cramer–Rao inequality, and a few other statistical concepts in the real domain to the corresponding complex domain. Optimizations of Hermitian forms under a linear constraint, a bilinear form under [...] Read more.

This paper deals with the extension of principal component analysis, canonical correlation analysis, the Cramer–Rao inequality, and a few other statistical concepts in the real domain to the corresponding complex domain. Optimizations of Hermitian forms under a linear constraint, a bilinear form under Hermitian-form constraints, and similar maxima/minima problems in the complex domain are discussed. Some vector/matrix differential operators are developed to handle the above types of problems. These operators in the complex domain and the optimization problems in the complex domain are believed to be new and novel. These operators will also be useful in maximum likelihood estimation problems, which will be illustrated in the concluding remarks. Detailed steps are given in the derivations so that the methods are easily accessible to everyone. Full article

(This article belongs to the Special Issue New Perspectives in Mathematical Statistics)

12 pages, 629 KiB

Open AccessArticle

An Efficient Iterative Approach for Hermitian Matrices Having a Fourth-Order Convergence Rate to Find the Geometric Mean

by Tao Liu, Ting Li, Malik Zaka Ullah, Abdullah Khamis Alzahrani and Stanford Shateyi

Mathematics 2024, 12(11), 1772; https://doi.org/10.3390/math12111772 - 6 Jun 2024

Cited by 1 | Viewed by 1275

Abstract

The target of this work is to present a multiplication-based iterative method for two Hermitian positive definite matrices to find the geometric mean. The method is constructed via the application of the matrix sign function. It is theoretically investigated that it has fourth [...] Read more.

The target of this work is to present a multiplication-based iterative method for two Hermitian positive definite matrices to find the geometric mean. The method is constructed via the application of the matrix sign function. It is theoretically investigated that it has fourth order of convergence. The type of convergence is also discussed, which is global under an appropriate choice of the initial matrix. Numerical experiments are reported based on input matrices of different sizes as well as various stopping termination levels with comparisons to methods of the same nature and same number of matrix–matrix multiplications. The simulation results confirm the efficiency of the proposed scheme in contrast to its competitors of the same nature. Full article

(This article belongs to the Special Issue Advances in Computational Mathematics and Applied Mathematics)

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