Stone's representation theorem of a group of random unitary operators on complete complex random inner product modules

2012 ◽  
Vol 42 (3) ◽  
pp. 181-202 ◽  
Author(s):  
TieXin GUO ◽  
Xia ZHANG
Keyword(s):  
Representation Theorem ◽  
Inner Product ◽  
Unitary Operators

scholarly journals Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 174
Author(s):  
Koen Thas
Keyword(s):  
Quantum Theory ◽  
Finite Fields ◽  
Hilbert Spaces ◽  
Inner Product ◽  
Unitary Operators ◽  
Evolution Operators ◽  
Quantum Theories ◽  
Classical Quantum ◽  
Absolute Quantum ◽  
Made In

In a recent paper, Chang et al. have proposed studying “quantum F u n ”: the q ↦ 1 limit of modal quantum theories over finite fields F q , motivated by the fact that such limit theories can be naturally interpreted in classical quantum theory. In this letter, we first make a number of rectifications of statements made in that paper. For instance, we show that quantum theory over F 1 does have a natural analogon of an inner product, and so orthogonality is a well-defined notion, contrary to what was claimed in Chang et al. Starting from that formalism, we introduce time evolution operators and observables in quantum F u n , and we determine the corresponding unitary group. Next, we obtain a typical no-cloning result in the general realm of quantum F u n . Finally, we obtain a no-deletion result as well. Remarkably, we show that we can perform quantum deletion by almost unitary operators, with a probability tending to 1. Although we develop the construction in quantum F u n , it is also valid in any other quantum theory (and thus also in classical quantum theory in complex Hilbert spaces).

On the Riesz-Fischer theorem

2004 ◽  
Vol 41 (4) ◽  
pp. 467-478
Author(s):  
J. Horváth
Keyword(s):  
Representation Theorem ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Riesz Representation Theorem ◽  
Riesz Representation

Are the two forms in which the theorem of the title is usually stated equivalent? We first summarize the three Comptes Rendus notes in which Frédéric Riesz published his results concerning L2, and then, in somewhat more detail, an article from 1910 which has been published only in Hungarian. Riesz deduces the two forms not from each other but both from the Fréchet—Riesz representation theorem. A theorem states that some of Riesz's results hold in the case of an abstract inner product space, and leads to maximal orthonormal systems which are not total. We conclude with a proof due to Ákos Császár which shows that a variant of Riesz's condition implies the Fischer form (i.e., completeness).

scholarly journals Spectral Theory of the Klein–Gordon Equation in Krein Spaces

2008 ◽  
Vol 51 (3) ◽  
pp. 711-750 ◽  
Author(s):  
Heinz Langer ◽  
Branko Najman ◽  
Christiane Tretter
Keyword(s):  
Gordon Equation ◽  
Main Tool ◽  
Inner Product ◽  
Continuous Group ◽  
Unitary Operators ◽  
Spectral Functions ◽  
Indefinite Inner Product ◽  
Krein Spaces ◽  
Klein Gordon ◽  
Klein Gordon Equation

AbstractIn this paper the spectral properties of the abstract Klein–Gordon equation are studied. The main tool is an indefinite inner product known as the charge inner product. Under certain assumptions on the potential V, two operators are associated with the Klein–Gordon equation and studied in Krein spaces generated by the charge inner product. It is shown that the operators are self-adjoint and definitizable in these Krein spaces. As a consequence, they possess spectral functions with singularities, their essential spectra are real with a gap around 0 and their non-real spectra consist of finitely many eigenvalues of finite algebraic multiplicity which are symmetric to the real axis. One of these operators generates a strongly continuous group of unitary operators in the Krein space; the other one gives rise to two bounded semi-groups. Finally, the results are applied to the Klein–Gordon equation in ℝn.

scholarly journals Inner Product Groups and Riesz Representation Theorem

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1946
Author(s):  
Alireza Pourmoslemi ◽  
Tahereh Nazari ◽  
Mehdi Salimi
Keyword(s):  
Representation Theorem ◽  
Abelian Groups ◽  
Inner Product ◽  
Riesz Representation Theorem ◽  
Product Group ◽  
Basic Properties ◽  
Inner Products ◽  
Normed Group ◽  
Riesz Representation ◽  
Torsion Free

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

scholarly journals General classes of impossible operations through the existence of incomparable states

2007 ◽  
Vol 7 (4) ◽  
pp. 392-400
Author(s):  
I. Chattopadhyay ◽  
D. Sarkar
Keyword(s):  
Large Class ◽  
General Class ◽  
Inner Product ◽  
Entangled States ◽  
Unitary Operators ◽  
Joint System ◽  
Bipartite State ◽  
Local Operation ◽  
Information Conservation ◽  
Hadamard Gate

In this work we show that the most general class of anti-unitary operators are nonphysical in nature through the existence of incomparable pure bipartite entangled states. It is also shown that a large class of inner-product-preserving operations defined only on the three qubits having spin-directions along x,y and z are impossible. If we perform such an operation locally on a particular pure bipartite state then it will exactly transform to another pure bipartite state that is incomparable with the original one. As subcases of the above results we find the nonphysical nature of universal exact flipping operation and existence of universal Hadamard gate. Beyond the information conservation in terms of entanglement, this work shows how an impossible local operation evolve with the joint system in a nonphysical way.

scholarly journals TEOREMA REPRESENTASI RIESZ–FRECHET PADA RUANG HILBERT

2011 ◽  
Vol 5 (2) ◽  
pp. 1-8
Author(s):  
Mozart W. Talakua ◽  
Stenly J. Nanuru
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Representation Theorem ◽  
Linear Functional ◽  
Inner Product ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Important Idea

Hilbert space is a very important idea of the Davids Hilbert invention. In 1907, Riesz and Fréchet developed one of the theorem in Hilbert space called the Riesz-Fréchet representationtheorem. This research contains some supporting definitions Banach space, pre-Hilbert spaces, Hilbert spaces, the duality of Banach and Riesz-Fréchet representation theorem. On Riesz-Fréchet representation theorem will be shown that a continuous linear functional that exist in the Hilbert space is an inner product, in other words, there is no continuous linear functional on a Hilbert space except the inner product.

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