Tensor Product Research Articles

Tensor Product Technique and Atomic Solution of Fractional Partial Differential Equations

In this paper, we investigate the atomic solution of a special type of fractional partial differential equations. Tensor product in Banach spaces, some properties of atom operators, and some properties of conformable fractional derivatives are utilized in such process.JEL Classification: 34G10, 34A55

Branching problem for tensoring two Verma modules and its application to differential symmetry breaking operators

Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of [Formula: see text] occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras [Formula: see text].

Long-time quantum scrambling and generalized tensor product structures

Long-time quantum scrambling and generalized tensor product structures

Tractability of Multivariate Approximation Problem on Euler and Wiener Integrated Processes

This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding to Euler and Wiener integrated processes with non-negative and nondecreasing smoothness parameters {rd}d∈N. We give matching sufficient and necessary conditions for various concepts of tractability in terms of the asymptotic properties of the regularity parameters, except for (s, 0)-WT.

THE SET OF ELEMENTARY TENSORS IS WEAKLY CLOSED IN PROJECTIVE TENSOR PRODUCTS

Abstract We prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we answer a question of Rodríguez and Rueda Zoca [‘Weak precompactness in projective tensor products’, Indag. Math. (N.S.)35(1) (2024), 60–75], proving that if $(x_n) \subset X$ and $(y_n) \subset Y$ are two weakly null sequences such that $(x_n \otimes y_n)$ converges weakly in $X \widehat {\otimes }_\pi Y$ , then $(x_n \otimes y_n)$ is also weakly null.

A-vertex magicness of product of graphs

In this paper, we discuss the A-vertex magicness of some products of graphs, where A is a non-trivial additive Abelian group with identity 0. We characterize A-vertex magicness of the Cartesian product of Pn and Pm , where A has at least three elements, and we also discuss the A -vertex magicness of co-normal and tensor product of two graphs. Finally, we give a new method to construct an infinite number of A-vertex magic graphs from existing ones using the join operation. As a consequence we obtain an earlier result in [Balamoorthy et al. in AKCE Int. J. Graphs Comb. (2022) 19 (3), 268–275]

Extremal marginals of an unbounded local completely positive and local completely contractive map

In this paper, we consider the unbounded local completely positive and local completely contractive maps on a maximal tensor product of unital locally C ∗ -algebras and discuss on extremal points of certain convex subsets in the set of such maps.

Antimagicness of Tensor product for some wheel related graphs with star

Antimagicness of Tensor product for some wheel related graphs with star

Stability verification of quantum non-i.i.d. sources

We introduce the problem of stability verification of quantum sources which are non-i.i.d.. The problem consists in ascertaining whether a given quantum source is stable or not, in the sense that it produces always a desired quantum state or if it suffers deviations. Stability is a statistical notion related to the sparsity of errors. This problem is closely related to the problem of quantum verification first proposed by Pallister et al (2018 Phys. Rev. Lett. 120), however, it extends the notion of the original problem. We introduce a family of states that come from these non-i.i.d. sources which we call a Markov state. These sources are more versatile than the i.i.d. ones as they allow statistical deviations from the norm instead of the more coarse previous approach. We prove in theorem 1 that the Markov states are not well described with tensor products over a changing source. In theorem 2 we further provide a lower bound on the trace distance between two Markov states, or conversely, an upper bound on the fidelity between these states. This is a bound on the capacity of determining the stability property of the source, which shows that it is exponentially easier to ascertain this with respect to n, the number of outcomes from the source.

Entanglement-assisted Quantum Reed-Muller Tensor Product Codes

We present the construction of standard entanglement-assisted (EA) qubit Reed-Muller (RM) codes and their tensor product variants from classical RM codes. We show that the EA RM codes obtained using the CSS construction have zero coding rate and negative catalytic rate. We further show that EA codes constructed from these same classical RM codes using the tensor product code (TPC) construction have positive coding rate and provide a subclass of EA RM TPCs that have positive catalytic rate, thus establishing the coding analog of superadditivity for this family of codes, useful towards quantum communications. We also generalize this analysis to obtain conditions for EA TPCs from classical codes to have positive catalytic rate when their corresponding EA CSS codes have zero rate.

Linear Combinations of Cluster Mean-Field States Applied to Spin Systems.

We present an innovative cluster-based method employing linear combinations of diverse cluster mean-field states and apply it to describe the ground state of strongly correlated spin systems. In cluster mean-field theory, the ground state wave function is expressed as a factorized tensor product of optimized cluster states. While our prior work concentrated on a single cMF tiling, this study removes that constraint by combining different tilings of cMF states. Selection criteria, including translational symmetry and spatial proximity, guide this process. We present benchmark calculations for the one- and two-dimensional J1 - J2 and XXZ Heisenberg models. Our findings highlight two key aspects. First, the method offers a semiquantitative description of the 0.4 ≲ J2/J1 ≲ 0.6 regime of the J1 - J2 model─a particularly challenging regime for existing methods. Second, our results demonstrate the capability of our method to provide qualitative descriptions for all the models and regimes considered, establishing it as a valuable reference. However, the inclusion of additional (weak) correlations is necessary for quantitative agreement, and we explore methods to incorporate these extra correlations.

Quantized Weyl algebras, the double centralizer property, and a new first fundamental theorem for Uq(gln)

Let P:=Pm×n denote the quantized coordinate ring of the space of m × n matrices, equipped with natural actions of the quantized enveloping algebras Uq(glm) and Uq(gln) . Let L and R denote the images of Uq(glm) and Uq(gln) in End(P) , respectively. We define a q-analogue of the algebra of polynomial-coefficient differential operators inside End(P) , henceforth denoted by PD , and we prove that L∩PD and R∩PD are mutual centralizers inside PD . Using this, we establish a new First Fundamental theorem of invariant theory for Uq(gln) . We also compute explicit formulas in terms of q-determinants for generators of the algebras Lh∩PD and Rh∩PD , where Lh and Rh denote the images of the Cartan subalgebras of Uq(glm) and Uq(gln) in End(P) , respectively. Our algebra PD and the algebra Pol(Matm,n)q that is defined in (Shklyarov et al 2004 Int. J. Math. 15 855–94) are related by extension of scalars, but we give a new construction of PD using deformed twisted tensor products.

Quadrotors’ double-loop controller design with tensor product model transformation and partial fully actuated method

In the existing work of tensor product (TP) model transformation, the TP model transformation-based work on the quadrotor’s control system design is scarce, the direct TP model transformation control strategy that applied to the quadrotor fails due to the calculation complexity, infeasibility of the huge amount of linear matrix inequalities, and the complexity of solving the linear matrix inequalities. To solve this problem, a partial TP model transformation-based double loop fuzzy controller has been studied in this work, the double-loop hybrid control scheme combines the fully actuated control method and the TP model transformation, while the fully actuated control method is used to the position subsystem control loop, and the TP model transformation based fuzzy controller is applied to the attitude control of the quadrotor. Moreover, for comparison purpose, a varying-input method based on TP model transformation is extended to the quadrotor’s system control. The double-loop hybrid control scheme could also be extended with other TP model transformation based tensor sampling methods, such as, uniform sampling method and varying-input method. At last, the proposed algorithms are evaluated and compared on Parrot Mambo Minidrone. Moreover, the varying-input control method is also evaluated on MFP450 and CUAV V5+ based hexarotor to verify the proposed method.

Simple modules for untwisted affine Lie algebras induced from nilpotent loop subalgebras

We construct large families of simple modules for untwisted affine Lie algebras using induction from one-dimensional modules over nilpotent loop subalgebras. We also show that the vector space of the first self-extensions for these module has uncountable dimension and that generic tensor products of these modules are simple.

Spatial Invariant Tensor Self-Representation Model for Hyperspectral Anomaly Detection.

With the development of hyperspectral imaging technology, the hyperspectral anomaly has attracted considerable attention due to its significant role in many applications. Hyperspectral images (HSIs) with two spatial dimensions and one spectral dimension are intrinsically three-order tensors. However, most of the existing anomaly detectors were designed after converting the 3-D HSI data into a matrix, which destroys the multidimension structure. To solve this problem, in this article, we propose a spatial invariant tensor self-representation (SITSR) hyperspectral anomaly detection algorithm, which is derived based on the tensor-tensor product (t-product) to preserve the multidimension structure and achieve a comprehensive description of the global correlation of HSIs. Specifically, we exploit the t-product to integrate spectral information and spatial information, and the background image of each band is represented as the sum of the t-product of all bands and their corresponding coefficients. Considering the directionality of the t-product, we utilize two tensor self-representation methods with different spatial modes to obtain a more balanced and informative model. To depict the global correlation of the background, we merge the unfolding matrices of two representative coefficients and constrain them to lie in a low-dimensional subspace. Moreover, the group sparsity of anomaly is characterized by l2.1.1 norm regularization to promote the separation of background and anomaly. Extensive experiments conducted on several real HSI datasets demonstrate the superiority of SITSR compared with state-of-the-art anomaly detectors.

Tucker Tensor Approach for Accelerating Fock Exchange Computations in a Real-Space Finite-Element Discretization of Generalized Kohn-Sham Density Functional Theory.

The evaluation of Fock exchange is often the computationally most expensive part of hybrid functional density functional theory calculations in a systematically improvable, complete basis. In this work, we employ a Tucker tensor based approach that substantially accelerates the evaluation of the action of Fock exchange by transforming three-dimensional convolutional integrals into a tensor product of one-dimensional convolution integrals. Our numerical implementation uses a parallelization strategy that balances the memory and communication bottlenecks, alongside overlapping compute and communication operations to enhance computational efficiency and parallel scalability. The accuracy and computational efficiency are demonstrated on various systems, including Pt clusters of various sizes and a TiO2 cluster with 3684 electrons.

Scalable computation of anisotropic vibrations for large macromolecular assemblies

The Normal Mode Analysis (NMA) is a standard approach to elucidate the anisotropic vibrations of macromolecules at their folded states, where low-frequency collective motions can reveal rearrangements of domains and changes in the exposed surface of macromolecules. Recent advances in structural biology have enabled the resolution of megascale macromolecules with millions of atoms. However, the calculation of their vibrational modes remains elusive due to the prohibitive cost associated with constructing and diagonalizing the underlying eigenproblem and the current approaches to NMA are not readily adaptable for efficient parallel computing on graphic processing unit (GPU). Here, we present eigenproblem construction and diagonalization approach that implements level-structure bandwidth-reducing algorithms to transform the sparse computation in NMA to a globally-sparse-yet-locally-dense computation, allowing batched tensor products to be most efficiently executed on GPU. We map, optimize, and compare several low-complexity Krylov-subspace eigensolvers, supplemented by techniques such as Chebyshev filtering, sum decomposition, external explicit deflation and shift-and-inverse, to allow fast GPU-resident calculations. The method allows accurate calculation of the first 1000 vibrational modes of some largest structures in PDB ( > 2.4 million atoms) at least 250 times faster than existing methods.

Unitary (semi)causal quantum-circuit representation of black hole evaporation

A general structure of unitary evolution (evaporation) of the black hole, respecting causality imposed by the event horizon (semicausality), has been derived and presented in the language of quantum circuits. The resulting consequences for the evolution of the corresponding entanglement entropy and the entropy curve have been determined. As an illustration of the general scheme, two families of qubit toy models have been discussed: tensor product models and controlled non-product models.

Support varieties without the tensor product property

AbstractWe show that over a perfect field, every non‐semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties.

Quantum image representations based on density matrices in open quantum systems

So far, research on quantum image representation has gone through more than 20 years. During this time, the quantum image representation models used have almost all been based on state vectors. However, in practical problems, the environment and the principal quantum system cannot be separated, and isolated quantum systems do not exist in principle. This case is often referred to as an open quantum system. In open quantum systems, many problems involve density matrices, such as the calculation of Von Neumann entropy, the quantization of coherence, and the operator-sum representations of quantum operations. Therefore, the existing quantum image representation models are only suitable for closed quantum systems. To this end, the paper proposes three models that can not only represent quantum images in an open quantum system but also decompose the evolution process of quantum images utilizing operator-sum decomposition. These three models are the representation model of quantum gray-scale images, the tensor product representation model of quantum color images, and the representation model of quantum color images based on mixed states in the Bloch sphere, respectively. All these image representation models have strong correlations among them and are very different from their classical analogues. Between them, the biggest difference is that the paper employs density matrices, inspired by incoherent-coherent states, to represent quantum images rather than classical state vectors. By means of one of the representation models proposed in the paper, we finally demonstrate the evolution process of the quantum image going through the amplitude damping channel.