Tensor Product Of Spaces Research Articles

Constant gap between conventional strategies and those based on C*-dynamics for self-embezzlement

We consider a bipartite transformation that we call self-embezzlement and use it to prove a constant gap between the capabilities of two models of quantum information: the conventional model, where bipartite systems are represented by tensor products of Hilbert spaces; and a natural model of quantum information processing for abstract states on C*-algebras, where joint systems are represented by tensor products of C*-algebras. We call this the C*-circuit model and show that it is a special case of the commuting-operator model (in that it can be translated into such a model). For the conventional model, we show that there exists a constant ϵ0>0 such that self-embezzlement cannot be achieved with precision parameter less than ϵ0 (i.e., the fidelity cannot be greater than 1−ϵ0); whereas, in the C*-circuit model---as well as in a commuting-operator model---the precision can be 0 (i.e., fidelity 1).Self-embezzlement is not a non-local game, hence our results do not impact the celebrated Connes Embedding conjecture. Instead, the significance of these results is to exhibit a reasonably natural quantum information processing problem for which there is a constant gap between the capabilities of the conventional Hilbert space model and the commuting-operator or C*-circuit model.

Solving the Bose-Hubbard model in new ways

We introduce a new method for analysing the Bose-Hubbard model for an array of bosons with nearest neighbor interactions. It is based on a number-theoretic implementation of the creation and annihilation operators that constitute the model. One of the advantages of this approach is that it facilitates accurate computations involving multi-particle states. In particular, we provide a rigorous computer assisted proof of quantum phase transitions in finite systems of this type. Furthermore, we investigate properties of the infinite array via harmonic analysis on the multiplicative group of positive rationals. This furnishes an isomorphism that recasts the underlying Fock space as an infinite tensor product of Hecke spaces, i.e., spaces of square-integrable periodic functions that are a superposition of non-negative frequency harmonics. Under this isomorphism, the number-theoretic creation and annihilation operators are mapped into the Kastrup model of the harmonic oscillator on the circle. It also enables us to highlight a kinship of the model at hand with an array of spin moments with a local anisotropy field. This identifies an interesting physical system that can be mapped into the model at hand.

Uniqueness of Hahn–Banach extensions and some of its variants

In this paper, we analyze the various strengthening and weakening of the uniqueness of the Hahn–Banach extension. In addition, we consider the case in which Y is an ideal of X. In this context, we study the property (U)/(SU)/(HB) and property (wU)/(k-U) for a subspace Y of a Banach space X. We obtain various new characterizations of these properties. We study different kinds of stabilities resulting from these properties in the tensor product spaces, spaces of Bochner integrable functions, and the higher duals of Banach spaces. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is established that a hyperplane in \(c_0\) has property (HB) if and only if it is an M-summand. It is observed that a finite-dimensional subspace Y has property (k-U) in \(c_0\), in addition to that if Y is an ideal, then \(Y^*\) is a k-strictly convex subspace of \(\ell _1\) for some natural k.

Atomic Solution of Poisson Type Equation

In this paper we find a certain solution of Poisson type fractional differential equation using theory of tensor product of Banach spaces.

Daugavet property in projective symmetric tensor products of Banach spaces

We show that all the symmetric projective tensor products of a Banach space X have the Daugavet property provided X has the Daugavet property and either X is an L_1-predual (i.e., X^{*} is isometric to an L_1-space) or X is a vector-valued L_1-space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise “localised” versions of the Daugavet property [i.e., Daugavet points and Delta-points introduced in Abrahamsen et al. (Proc Edinb Math Soc 63:475–496 2020)] for L_1-preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real L_1-preduals in terms of the absence of Delta-points and also to provide new examples of L_1-preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces [in the sense of Werner (J Funct Anal 143:117–128, 1997)] so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for L_1-preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in Rueda Zoca (J Inst Math Jussieu 20(4):1409–1428, 2021) about the Daugavet property for projective tensor products is also obtained.

Continuous frames in tensor product Hilbert spaces, localization operators and density operators

Continuous frames and tensor products are important topics in theoretical physics. This paper combines those concepts. We derive fundamental properties of continuous frames for tensor product of Hilbert spaces. This includes, for example, the consistency property, i.e. preservation of the frame property under the tensor product, and the description of the canonical dual tensors by those on the Hilbert space level. We show the full characterization of all dual systems for a given continuous frame, a result interesting by itself, and apply this to dual tensor frames. Furthermore, we discuss the existence on non-simple tensor product (dual) frames. Continuous frame multipliers and their Schatten class properties are considered in the context of tensor products. In particular, we give sufficient conditions for obtaining partial trace multipliers of the same form, which is illustrated with examples related to short-time Fourier transform and wavelet localization operators. As an application, we offer an interpretation of a class of tensor product continuous frame multipliers as density operators for bipartite quantum states, and show how their structure can be restricted to the corresponding partial traces.

Countable tensor products of Hermite spaces and spaces of Gaussian kernels

In recent years finite tensor products of reproducing kernel Hilbert spaces (RKHSs) of Gaussian kernels on the one hand and of Hermite spaces on the other hand have been considered in tractability analysis of multivariate problems. In the present paper we study countably infinite tensor products for both types of spaces. We show that the incomplete tensor product in the sense of von Neumann may be identified with an RKHS whose domain is a proper subset of the sequence space RN. Moreover, we show that each tensor product of spaces of Gaussian kernels having square-summable shape parameters is isometrically isomorphic to a tensor product of Hermite spaces; the corresponding isomorphism is given explicitly, respects point evaluations, and is also an L2-isometry. This result directly transfers to the case of finite tensor products. Furthermore, we provide regularity results for Hermite spaces of functions of a single variable.

Dominated multilinear operators defined on tensor products of Banach spaces

Dominated multilinear operators defined on tensor products of Banach spaces

Generalized Fusion Frame in A Tensor Product of Hilbert Space

Generalized fusion frames and some of their properties in a tensor product of Hilbert spaces are studied. Also, the canonical dual g-fusion frame in a tensor product of Hilbert spaces is considered. The frame operator for a pair of g-fusion Bessel sequences in a tensor product of Hilbert spaces is presented.

Norm-Attaining Tensors and Nuclear Operators

Given two Banach spaces X and Y, we introduce and study a concept of norm-attainment in the space of nuclear operators \(\mathcal N(X,Y)\) and in the projective tensor product space \(X\widehat{\otimes }_\pi Y\). We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in \(\mathcal N(X,Y)\) and in \(X\widehat{\otimes }_\pi Y\) is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in \(X\widehat{\otimes }_\pi Y\) which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.

О РЕАЛИЗАЦИИ ПРИНЦИПА СУПЕРПОЗИЦИИ ДЛЯ КОНЕЧНОГО ПУЧКА ИНТЕГРАЛЬНЫХ КРИВЫХ БИЛИНЕЙНОЙ СИСТЕМЫВТОРОГОПОРЯДКА. I

Based on the projectivization of the non-linear Rayleigh-Ritz functional operator and the tensor product of real Hilbert spaces, for a finite bundle of integral curves of a controlled bilinear system of the second order, necessary and sufficient conditions for the existence of a differential realization of this bundle in the class of linear non-stationary ordinary differential equations of the second order are determined (including hyperbolic models) in a separable Hilbert space. In this case, the original bilinear structure models the nonlinearity of the system dynamics, both the trajectory itself and the speed of movement on this trajectory. The results obtained have applications in the theory of inverse problems of non-stationary controlled multilinear differential models of higher orders and the theory of optimal control using the technology of successive approximations in solving a two-point boundary value problem based on the quasi-linearization procedure using the Picard method.

Spectra of the Energy Operator of Two-Electron System in the Impurity Hubbard Model

We consider two-electron systems for the impurity Hubbard Model and investigate the spectrum of the system in a singlet state for the v-dimensional integer valued lattice Zv. We proved the essential spectrum of the system in the singlet state is consists of union of no more then three intervals, and the discrete spectrum of the system in the singlet state is consists of no more then five eigenvalues. We show that the discrete spectrum of the system in the triplet and singlet states differ from each other. In the singlet state the appear additional two eigenvalues. In the triplet state the discrete spectrum of the system can be empty set, or is consists of one-eigenvalue, or is consists of two eigenvalues, or is consists of three eigenvalues. For investigation the structure of essential spectra and discrete spectrum of the energy operator of two-electron systems in an impurity Hubbard model, for which the momentum representation is convenient. In addition, we used the tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces and described the structure of essential spectrum and discrete spectrum of the energy operator of two-electron systems in an impurity Hubbard model.

Controlled generalized fusion frame in the tensor product of Hilbert spaces

We present controlled by operators generalized fusion frame in the tensor product of Hilbert spaces and discuss some of its properties. We also describe the frame operator for a pair of controlled $g$-fusion Bessel sequences in the tensor product of Hilbert spaces.

Tensor polynomial identities

Tensor polynomial identities generalize the concept of polynomial identities on $d \times d$ matrices to identities on tensor product spaces. Here we completely characterize a certain class of tensor polynomial identities in terms of their associated Young tableaux. Furthermore, we provide a method to evaluate arbitrary alternating tensor polynomials in $d^2$ variables.

Quantum mereology in finite quantum mechanics

Any Hilbert space with composite dimension can be factored into a tensor product of smaller Hilbert spaces. This allows us to decompose a quantum system into subsystems. We propose a model based on finite quantum mechanics for a constructive study of such decompositions.

Quantum Electrostatics, Gauss’s Law, and a Product Picture for Quantum Electrodynamics; or, the Temporal Gauge Revised

We provide a theoretical foundation for the notion of the quantum coherent state of the electrostatic field of a static external charge distribution introduced in a 1998 paper and rederive formulae there for the inner products of a pair of such states. Contrary to what one might expect, these inner products are non-zero whenever the total charges of the two charge distributions are equal, even if the charge distributions themselves differ. We actually display two different frameworks for these same coherent states, in the second of which Gauss's law only holds in expectation value. We propose an experiment capable of ruling that out. The first framework leads to a 'product picture' for full QED -- i.e. a reformulation of standard QED in which it has a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free charged-matter Hamiltonian and an interaction term, acting on a 'physical subspace' of the full tensor product of charged-matter and electromagnetic-field Hilbert spaces. (The traditional Coulomb gauge formulation of QED isn't a product picture because, in it, the longitudinal part of the electric field is a function of the charged matter operators.) We do this for both Maxwell-Dirac and Maxwell-Schr\"odinger QED. For all states in the physical subspace of each of these systems, the charged matter is entangled with longitudinal photons and Gauss's law holds on the physical subspace as an operator equation; albeit the electric field operator and the Hamiltonian, while self-adjoint on the physical subspace, fail to be self-adjoint on the full tensor-product Hilbert space. Analogues of our coherent state inner products and of the product picture play a role in the author's matter-gravity entanglement hypothesis. Also, the product picture amounts to a temporal gauge quantization of QED which appears to be free from the difficulties of previous versions.

The operator system of Toeplitz matrices

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of n × n n\times n Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than n n . The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the n × n n\times n complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the n × n n\times n Toeplitz matrices into the algebra of all n × n n\times n complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of n × n n\times n complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix ξ n \xi _n generates an extremal ray in the cone of all continuous n × n n\times n Toeplitz-matrix valued functions f f on the unit circle S 1 S^1 whose Fourier coefficients f ^ ( k ) \hat f(k) vanish for | k | ≥ n |k|\geq n . Lastly, it is noted that all positive Toeplitz matrices over nuclear C ∗ ^* -algebras are approximately separable.

A characterization of Riesz bases and pair of dual frames in tensor product of Hilbert spaces

A characterization of Riesz bases and pair of dual frames in tensor product of Hilbert spaces

Applications of Multiparameter Eigenvalue Problems

It was mainly due to Atkinson works, who introduced Linear Multiparameter Eigenvalue problems (LMEPs), based on determinantal operators on the Tensor Product Space. Later, in the area of Multiparameter eigenvalue problems has received attention from the Mathematicians in the recent years also, who pointed out that there exist a variety of mixed eigenvalue problems with several parameters in different scientific domains. This article aims to bring into a light variety of scientific problems that appear naturally as LMEPs. Of course, with all certainty, the list of collection of applications presented here are far from complete, and there are bound to be many more applications of which we are currently unaware. The paper may provide a review on applications of Multiparameter eigenvalue problems in different scientific domains and future possible applicatios both in theoretical and applied disciplines.

Solutions of Certain Fractional Partial Differential Equations

In this paper we find certain solutions of some fractional partial differential equations. Tensor product of Banach spaces is used where separation of variables does not work.