Entanglement of Multipartite Fermionic Coherent States for Pseudo-Hermitian Hamiltonians

We study the entanglement of multiqubit fermionic pseudo-Hermitian coherent states (FPHCSs) described by anticommutative Grassmann numbers. We introduce pseudo-Hermitian versions of well-known maximally entangled pure states, such as Bell, GHZ, Werner, and biseparable states, by integrating over the tensor products of FPHCSs with a suitable choice of Grassmannian weight functions. As an illustration, we apply the proposed method to the tensor product of two- and three-qubit pseudo-Hermitian systems. For a quantitative characteristic of entanglement of such states, we use a measure of entanglement determined by the corresponding concurrence function and average entropy.     

An integral relation for tensor polynomials

We prove two lemmas and one theorem that allow integrating the product of an arbitrary number of unit vectors and the Legendre polynomials over a sphere of arbitrary radius. Such integral tensor products appear in solving inhomogeneous Helmholtz equations whose right-hand side is proportional to the product of a nonfixed number of unit vectors.     

A tensor product matrix approximation problem in quantum physics

We consider a matrix approximation problem arising in the study of entanglement in quantum physics. This notion represents a certain type of correlations between subsystems in a composite quantum system. The states of a system are described by a density matrix, which is a positive semidefinite matrix with trace one. The goal is to approximate such a given density matrix by a so-called separable density matrix, and the distance between these matrices gives information about the degree of entanglement in the system. Separability here is expressed in terms of tensor products. We discuss this approximation problem for a composite system with two subsystems and show that it can be written as a convex optimization problem with special structure. We investigate related convex sets, and suggest an algorithm for this approximation problem which exploits the tensor product structure in certain subproblems. Finally some computational results and experiences are presented.     

Atomic decompositions for tensor products and polynomial spaces

We study the existence of atomic decompositions for tensor products of Banach spaces and spaces of homogeneous polynomials. If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product , for any symmetric tensor norm μ. In addition, the reciprocal statement is investigated and analogous consequences for the full tensor product are obtained. Finally we apply the previous results to establish the existence of monomial atomic decompositions for certain ideals of polynomials on X.     

Transformations on tensor spaces II

We prove that a linear map of one tensor product space to another sending decomposable tensors to decomposable tensors is essentially a tensor product of linear maps of products of component factors of the domain into a selection of the factors of the range. The product of those factors of the domain not involved in the above is collapsed via a linear functional, and those factors in the range left out of the above provide a common factor in the range. In the statement of the main theorem the flanking maps are induced by permutations of the factors of the domain and the range and they present the products in manageable form. It is assumed that the underlying field has at least five members, but the necessity of this assumption is not settled. All vector spaces are finite dimensional and the tensor products have finitely many components.     

Linear invariants of a Cartesian tensor

The number of linear invariants under SO(3) as well as SO(2)of a Cartesian tensor of an arbitrary rank is studied. A linearform is defined in terms of elements of a tensor. It is establishedthat the number of linear invariants of a tensor of rank n underSO(3) equals the dimension of the space of isotropic tensorsof rank n. Formulas for the number of invariants in the twocases are also derived. For the elasticity tensor, our analysisconfirms the results of Norris.     

Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products

We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.     

Lattice tensor products. IV

In 1999, for lattices A and B, G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B. In Part I of this paper, we showed that for a finite lattice A and a bounded lattice B, this construction can be "coordinatized,'' that is, represented in B ^A so that the representing elements are easy to recognize. In this note, we show how to extend our method to an arbitrary bounded lattice A to coordinatize A?B.     

Infinite tensor products of Banach spaces I

In this paper we construct a new class of infinite tensor product Banach spaces. We call them spaces of type v since they are constructed in a manner which closely parallels von Neumanns construction of infinite tensor products of Hilbert spaces. We use these spaces to present results on infinite tensor products of semigroups of operators.     

Average case tractability of non-homogeneous tensor product problems with the absolute error criterion

We study average case tractability of non-homogeneous tensor product problems with the absolute error criterion. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. For general non-homogeneous tensor product problems, we obtain the matching necessary and sufficient conditions for strong polynomial tractability in terms of the one-dimensional eigenvalues. We give some examples to show that strong polynomial tractability is not equivalent to polynomial tractability, and polynomial tractability is not equivalent to quasi-polynomial tractability. But for non-homogeneous tensor product problems with decreasing eigenvalues, we prove that strong polynomial tractability is always equivalent to polynomial tractability, and strong polynomial tractability is even equivalent to quasi-polynomial tractability when the one-dimensional largest eigenvalues are less than one. In particular, we find an example that quasi-polynomial tractability with the absolute error criterion is not equivalent to that with the normalized error criterion even if all the one-dimensional largest eigenvalues are one. Finally we consider a special class of non-homogeneous tensor product problems with improved monotonicity condition of the eigenvalues.     

A note on linear preservers of certain ranks of tensor products of matrices

In this note, we give a simple proof as well as an extension of a very recent result of B. Zheng, J. Xu and A. Fosner concerning linear maps between vector spaces of complex square matrices that preserve the rank of tensor products of matrices by using a structure theorem of R. Westwick on linear maps between tensor product spaces that preserve non-zero decomposable elements.     

On the properties of a new tensor product of matrices

Previously, the author introduced a new tensor product of matrices according to which the matrix of the discrete Walsh-Paley transform can be represented as a power of the second-order discrete Walsh transform matrix H with respect to this product. This power is an analogue of the representation of the Sylvester-Hadamard matrix in the form of a Kronecker power of H. The properties of the new tensor product of matrices are examined and compared with those of the Kronecker product. An algebraic structure with the matrix H used as a generator element and with these two tensor products of matrices is constructed and analyzed. It is shown that the new tensor product operation proposed can be treated as a convenient mathematical language for describing the foundations of discrete Fourier analysis.     

Locally tensor product functions

We present in this paper a family of functions which are tensor product functions in subdomains, while not having the usual drawback of functions which are tensor product functions in the whole domain. With these functions we can add more points in some region without adding points on lines parallel to the axes. These functions are linear combinations of tensor product polynomial B-splines, and the knots of different B-splines are less connected together than with usual polynomial B-splines. Approximation of functions, or data, with such functions gives satisfactory results, as shown by numerical experimentation. AMS subject classification 41A15, 41A63, 65Dxx     

A theory of tensor products for modulé categories for a vertex operator algebra, IV

This is the fourth part in a series of papers developing a tensor product theory of modules for a vertex operator algebra. In this paper, we establish the associativity of P(z)-tensor products for nonzero complex numbers z constructed in Part III of the present series under suitable conditions. The associativity isomorphisms constructed in this paper are analogous to associativity isomorphisms for vector space tensor products in the sense that they relate the tensor products of three elements in three modules taken in different ways. The main new feature is that they are controlled by the decompositions of certain spheres with four punctures into spheres with three punctures using a sewing operation. We also show that under certain conditions, the existence of the associativity isomorphisms is equivalent to the associativity (or (nonmeromorphic) operator product expansion in the language of physicists) for the intertwining operators (or chiral vertex operators). Thus the associativity of tensor products provides a means to establish the (nonmeromorphic) operator product expansion.     

Singular values of a real rectangular tensor

Real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we systematically study properties of singular values of a real rectangular tensor, and give an algorithm to find the largest singular value of a nonnegative rectangular tensor. Numerical results show that the algorithm is efficient.     

有限自动机的积与路代数的张量积

讨论了两个有限自动机积的路代数,得出了这些积的路代数与这两个有限自动机路代数的张量积的一些关系.     

Unique tensor factorization of algebras

Tensor product decomposition of algebras is known to be non-unique in many cases. But, as will be shown here, a -indecomposable, finite-dimensional -algebra A has an essentially unique tensor factorization into non-trivial, -indecomposable factors . Thus the semiring of isomorphism classes of finite-dimensional -algebras is a polynomial semiring . Moreover, the field of complex numbers can be replaced by an arbitrary field of characteristic zero if one restricts oneself to schurian algebras. Received: 5 October 1998     

Note on the wave equation and tensor products

In this study we apply convolution and tensor products of distribution to solve the non-homogenous wave equation with initial condition and discuss the uniqueness and continuity of solution. We also show that the tensor product can be applied to compute the some singular integrals.