Tensor products and probability weights

We study a general tensor product for two collections of related physical operations or observations. This is a free product, subject only to the condition that the operations in the first collection fail to have any influence on the statistics of operations in the second collection and vice versa. In the finite-dimensional case, it is shown that the vector space generated by the probability weights on the general tensor product is the algebraic tensor product of the vector spaces generated by the probability weights on the components. The relationship between the general tensor product and the tensor product of Hilbert spaces is examined in the light of this result.     

Tensor products in generalized measure theory

Kläy, Randall, and Foulis established that the signed weight space of the tensor product of two quasimanuals each having a positive, finite-dimensional state space is isomorphic to the algebraic tensor product of the signed-weight spaces of the factors. We obtain a generalization of this result for arbitrary quasimanuals. A compactness condition due to Cook—here calleddiscreteness—is discussed and shown to be preserved under the formation of tensor products. It is shown that the predual of the signed weight space of a tensor product of discrete manuals is the projective (ordered) tensor product of the preduals of the signed weight spaces of the factors.     

Tensor product of difference posets and effect algebras

A tensor product of difference posets and/or, equivalently, of effect algebras, which generalize orthoalgebras and orthomodular posets, is defined, and an equivalent condition is presented. The proof uses the notion of D-test spaces generalizing test spaces of Randall and Foulis. In particular, we show that a tensor product for difference posets with a nonempty system of probability measures exists.     

An application of ideal experiments to quantum mechanical measurement theory

It is shown that the rule for obtaining probabilities by squaring amplitudes is deducible from ideal experiments in a mechanics of unitary motion in a complex-linear space, with tensor product for making compound systems. Difficulty with tensor product in undoctored quaternionic quantum mechanics makes the argument inapplicable there. Except for the replacement by ideal experiments of a more formal unitary equivalence, the discussion is similar to that of Everett (1957). Diagonal expression of a general vector in the tensor product of two spaces is related to polar form of a matrix, in the Appendix.     

Tensor product of frame manuals

Since its first use, there has been widespread dissatisfaction with the Hilbert-space tensor product as a device for coupling the Hilbert-space models of two separated quantum mechanical systems. The Hilbert-space model is paraphrased manual-theoretically by the assertion that quantum mechanical entities are represented by frame manuals. There is a natural, heuristically straightforward tensor product for (unital) manuals, and it is natural therefore to ask whether the tensor product of frame manuals might serve as an alternative model of separated quantum mechanical systems. It is shown that the states on a tensor product of complex frame manuals give rise uniquely to sesquilinear forms on the tensor product of the underlying Hibert spaces. In certain cases, these in turn give rise to operators, which, however, are not generally positive, and which, even if compact, need not be trace-class.     

Tensor products of Hilbert space effect algebras

A definition of a tensor product in the category of Hilbert space effect algebras is introduced such that the tensor product reflects as much as possible of the physically important properties of the components. It is shown that in the complex case, there are two candidates to the tensor product, which are not equivalent. The situation is similar to the tensor product in the category of projection lattices, but also the two dimensional case is included. The case of tensor product of two classical systems is also investigated.     

The tensor product of generalized sample spaces which admit a unital set of dispersion-free weights

Techniques for constructing the tensor product of two generalized sample spaces which admit unital sets of dispersion-free weights are discussed. A duality theory is developed, based on the 1-cuts of the dispersion-free weights, and used to produce a candidate for the tensor product. This construction is verified for Dacification manuals, a conjecture is given for other reflexive cases, and some adjustments for nonreflexive cases are considered. An alternate approach, using graphs of interpretation morphisms on the duals, is also presented.     

Partial isometries and quotient structures in Hilbert spaces

Partial isometries are studied as the natural framework both for the representation of semi-groups on Hilbert spaces and for the mapping of operators with different spectra. The general theory is illustrated by examining several pertinent problems from conventional quantum mechanics. Families of partial isometries are found to induce quotient structures on Hilbert space. Embedding in appropriate tensor product spaces allows the representation of such families by a single isometry.     

Direct product and decomposition of certain physically important algebras

I consider the direct product algebra formed from two isomorphic Clifford algebras. More specifically, for an element x in each of the two component algebras I consider elements in the direct product space with the form x x. I show how this construction can be used to model the algebraic structure of particular vector spaces with metric, to describe the relationship between wavefunction and observable in examples from quantum mechanics, and to express the relationship between the electromagnetic field tensor and the stress-energy tensor in electromagnetism. To enable this analysis I introduce a particular decomposition of the direct product algebra.     

Algebraic structure of general electromagnetic fields and energy flow

The algebraic structures of a general electromagnetic field and its energy–momentum tensor in a stationary space–time are analyzed. The explicit form of the reference frame in which the energy of the field appears at rest is obtained in terms of the eigenvectors of the electromagnetic tensor and the existing Killing vector. The case of a stationary electromagnetic field is also studied and a comparison is made with the standard short-wave approximation. The results can be applied to the general case of a structured light beams, in flat or curved spaces. Bessel beams are worked out as example.     

Bipartite subspaces having no bases distinguishable by local operations and classical communication

It is proved that there exist subspaces of bipartite tensor product spaces that have no orthonormal bases that can be perfectly distinguished by means of local operations and classical communication. A corollary of this fact is that there exist quantum channels having suboptimal classical capacity even when the receiver may communicate classically with a third party that represents the channel's environment.     

Decoupling braided tensor factors

We briefly report on our result that the braided tensor product algebra of two module algebras A ₁, A ₂ of a quasitriangular Hopf algebra H is equal to the ordinary tensor product algebra of A ₁ with a subalgebra isomorphic to A ₂ and commuting with A ₁, provided there exists a realization of H within A ₁. As applications of the theorem, we consider the braided tensor product algebras of two or more quantum group covariant quantum spaces or deformed Heisenberg algebras.     

On positive maps in quantum information

Using the Grothendieck approach to the tensor product of locally convex spaces, we review a characterization of positive maps as well as Belavkin-Ohya characterization of PPT states. Moreover, within this scheme, a generalization of the idea of Choi matrices for genuine quantum systems will be presented.     

Commutativity equations and dressing transformations

We study dressing transformations that generate all solutions to commutativity equations and, after picking up special coordinates, all solutions to WDVV equations. We conjecture that the homological tensor product of solutions to the commutativity equations corresponds to the tensor product of matrices of the dressing transformation and check this in the first nontrivial case.     

Decomposition of the space of covariant two-tensors on R3

We show that the decomposition of the space of covariant two-tensors onR ³ is true in weighted Hölderian spaces, as in weighted Sobolev spaces, in the general case, that is without supposing the metric near the flat metric. M. Cantor proved, first, that a splitting of two-covariant tensor fields onR ⁿ in weighted Sobolev spaces was true. We apply this result to solve the problem of constraints, in general relativity; we show that this problem admits a solution in the most general case.     

On the curvature of warped product Finsler spaces and the Laplacian of the Sasaki–Finsler metrics

As an opening, we prove that a warped product Finsler space F=_F1×fF2$F=F_1{\times }_f{F}_2$ is of constant curvature c$c$ if and only if the base space _F1$F_1$ is also of constant curvature c$c$, the fiber space _F2$F_2$ is of some constant curvature α$\alpha$, and five other partial differential equations are satisfied. A rather similar result is proved for the case of warped product Finsler spaces of scalar curvature. Close relationships between the geometry of the warped product Finsler spaces of constant curvature and the spectral theory of the Laplacian (Laplace–Beltrami operator) of the well-known Sasaki–Finsler metrics of the base space _F1$F_1$ is established by detailed investigation of the above mentioned PDEs. We also define a new tensor for warped product Finsler spaces, which we call a warped-Cartan tensor. Using the tensor we define a new class of warped product Finsler spaces, calling them C-Warped spaces, which contain Landsberg, Berwald, locally Minkowski and Riemannian spaces, but not necessarily all of the constant curvature Finsler spaces of warped product type. Several results are obtained and special cases, for example the case of Riemannian, C-Warped and projectively flat spaces are also considered.     

Construction of non-Abelian gauge theories on noncommutative spaces

We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with a constant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories. Received: 13 June 2001 / Published online: 19 July 2001     

一种彩色图像边缘检测方法

研究了基于小波的彩色图像多尺度边缘检测的方法。选择了感知性强的YUV色彩空间，根据小波理论用多分辨分析的张量积代替小波函数的张量积，建立了二维的Mallat算法以实现快速小波变换。并对1个标准图样进行了多尺度的小波分解，得到各尺度下的边缘检测结果。可以看出，在较高尺度和较低尺度检测时，实验结果与理论分析相吻合，效果较好。     

Homological equations for tensor fields and periodic averaging

Homological equations of tensor type associated to periodic flows on a manifold are studied. The Cushman intrinsic formula [4] is generalized to the case of multivector fields and differential forms. Some applications to normal forms and the averaging method for perturbed Hamiltonian systems on slow-fast phase spaces are given.