Topic
Product (mathematics)
About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.
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TL;DR: In this article, the authors describe stabilizer states and Clifford group operations using linear operations and quadratic forms over binary vector spaces, and show how the n-qubit Clifford group is isomorphic to a group with an operation defined in terms of a $(2n+1)$ binary matrix product.
Abstract: We describe stabilizer states and Clifford group operations using linear operations and quadratic forms over binary vector spaces. We show how the n-qubit Clifford group is isomorphic to a group with an operation that is defined in terms of a $(2n+1)\ifmmode\times\else\texttimes\fi{}(2n+1)$ binary matrix product and binary quadratic forms. As an application we give two schemes to efficiently decompose Clifford group operations into one- and two-qubit operations. We also show how the coefficients of stabilizer states and Clifford group operations in a standard basis expansion can be described by binary quadratic forms. Our results are useful for quantum error correction, entanglement distillation, and possibly quantum computing.
171 citations
TL;DR: This article obtained Zakrzewski's deformation of Fun SL(2) through the construction of a *-product on SL (2) and then gave the deformation for both algebras.
Abstract: We obtain Zakrzewski's deformation of Fun SL(2) through the construction of a *-product on SL(2). We then give the deformation of
$$(\mathfrak{s}\mathfrak{l}({\text{2}}))$$
dual to this, as well as a Poincare basis for both algebras.
170 citations
TL;DR: In this paper, the warped product of the shape space and the half-line of the Riemannian metric space was studied for the general case where k = 1, k = 2 and k = 3.
Abstract: The Riemannian metric structure of the shape space $\sum^k_m$ for $k$ labelled points in $\mathbb{R}^m$ was given by Kendall for the atypically simple situations in which $m = 1$ or 2 and $k \geq 2$. Here we deal with the general case $(m \geq 1, k \geq 2)$ by using the properties of Riemannian submersions and warped products as studied by O'Neill. The approach is via the associated size-and-shape space that is the warped product of the shape space and the half-line $\mathbb{R}_+$ (carrying size), the warping function being equal to the square of the size. When combined with parallel studies by Le of the corresponding global geodesic geometry, the results obtained here determine the environment in which shape-statistical calculations have to be acted out. Finally three different applications are discussed that illustrate the theory and its use in practice.
170 citations
TL;DR: In this article, the closed-form solution of the equations of motion of an ideal missile pursuing a nonmaneuvering target according to the pure proportional navigation law is obtained as a function of the polar coordinates for all real navigation constants N>or=2.
Abstract: The closed-form solution of the equations of motion of an ideal missile pursuing a nonmaneuvering target according to the pure proportional navigation law is obtained as a function of the polar coordinates for all real navigation constants N>or=2. The solution is given in the form of a uniformly convergent infinite product which reduces to a product of a finite number of factors if the navigation constant is a rational number. The solution is discussed, and necessary and sufficient conditions are stated for vanishing, bounded, and unbounded missile acceleration in the final phase of pursuit. >
170 citations
TL;DR: In this article, the concept of semi-direct product for Hopf algebras has been investigated, and two notions of coadjoint action and smash coproduct have been introduced: the well-known concept of smash product and the dual notion of smash co-product.
170 citations