CANONICAL REPRESENTATION OF 3-QUBIT STATES WITH REAL AMPLITUDES

AbstractSome basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.

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Practical Tools for Reasoning About Linear Constraints

Fundamenta Informaticae ◽

10.3233/fi-1991-153-410 ◽

1991 ◽

Vol 15 (3-4) ◽

pp. 357-379

Author(s):

Tien Huynh ◽

Leo Joskowicz ◽

Catherine Lassez ◽

Jean-Louis Lassez

Keyword(s):

Logic Programming ◽

Intelligent Systems ◽

Optimization Problem ◽

Spatial Reasoning ◽

Quantifier Elimination ◽

Canonical Representation ◽

Linear Constraints ◽

Practical Applications ◽

Variable Elimination ◽

Fourier Algorithm

We address the problem of building intelligent systems to reason about linear arithmetic constraints. We develop, along the lines of Logic Programming, a unifying framework based on the concept of Parametric Queries and a quasi-dual generalization of the classical Linear Programming optimization problem. Variable (quantifier) elimination is the key underlying operation which provides an oracle to answer all queries and plays a role similar to Resolution in Logic Programming. We discuss three methods for variable elimination, compare their feasibility, and establish their applicability. We then address practical issues of solvability and canonical representation, as well as dynamical updates and feedback. In particular, we show how the quasi-dual formulation can be used to achieve the discriminating characteristics of the classical Fourier algorithm regarding solvability, detection of implicit equalities and, in case of unsolvability, the detection of minimal unsolvable subsets. We illustrate the relevance of our approach with examples from the domain of spatial reasoning and demonstrate its viability with empirical results from two practical applications: computation of canonical forms and convex hull construction.

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PROPERTIES OF QUASI-RELABELING TREE BIMORPHISMS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054110007234 ◽

2010 ◽

Vol 21 (03) ◽

pp. 257-276 ◽

Cited By ~ 1

Author(s):

ANDREAS MALETTI ◽

CĂTĂLIN IONUŢ TÎRNĂUCĂ

Keyword(s):

Cartesian Product ◽

Canonical Representation ◽

Finite Union ◽

Top Down ◽

Tree Language ◽

Regular Tree ◽

Fundamental Properties ◽

Tree Transducers ◽

Tree Languages ◽

Context Free

The fundamental properties of the class QUASI of quasi-relabeling relations are investigated. A quasi-relabeling relation is a tree relation that is defined by a tree bimorphism (φ, L, ψ), where φ and ψ are quasi-relabeling tree homomorphisms and L is a regular tree language. Such relations admit a canonical representation, which immediately also yields that QUASI is closed under finite union. However, QUASI is not closed under intersection and complement. In addition, many standard relations on trees (e.g., branches, subtrees, v-product, v-quotient, and f-top-catenation) are not quasi-relabeling relations. If quasi-relabeling relations are considered as string relations (by taking the yields of the trees), then every Cartesian product of two context-free string languages is a quasi-relabeling relation. Finally, the connections between quasi-relabeling relations, alphabetic relations, and classes of tree relations defined by several types of top-down tree transducers are presented. These connections yield that quasi-relabeling relations preserve the regular and algebraic tree languages.

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