Strongly Irreducible Operators and Indecomposable Representations of Quivers on Infinite-Dimensional Hilbert Spaces

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

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Isometric Group Actions on Hilbert Spaces: Structure of Orbits

Canadian Journal of Mathematics ◽

10.4153/cjm-2008-044-4 ◽

2008 ◽

Vol 60 (5) ◽

pp. 1001-1009 ◽

Cited By ~ 2

Author(s):

Yves de Cornulier ◽

Romain Tessera ◽

Alain Valette

Keyword(s):

Hilbert Space ◽

Nilpotent Group ◽

Hilbert Spaces ◽

Metabelian Group ◽

Isometric Action ◽

Finitely Generated ◽

Infinite Dimensional ◽

Infinite Dimensional Hilbert Space ◽

Isometric Group Actions ◽

Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

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Elements of Spectral Theory for Generalized Derivations II : The Semifredholm Domain

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-091-4 ◽

1981 ◽

Vol 33 (5) ◽

pp. 1205-1231 ◽

Cited By ~ 2

Author(s):

Lawrence A. Fialkow

Keyword(s):

Hilbert Spaces ◽

Simple Formula ◽

Decomposition Theorem ◽

Linear Operators ◽

Generalized Derivations ◽

Bounded Linear ◽

Infinite Dimensional ◽

Hilbert Space Operators ◽

Algebraic Invariants ◽

Fredholm Theorem

Let and denote infinite dimensional Hilbert spaces and let denote the space of all bounded linear operators from to . For A in and B in , let τAB denote the operator on defined by τAB(X) = AX – XB. The purpose of this note is to characterize the semi-Fredholm domain of τAB (Corollary 3.16). Section 3 also contains formulas for ind(τAB – λ). These results depend in part on a decomposition theorem for Hilbert space operators corresponding to certain “singular points” of the semi-Fredholm domain (Theorem 2.2). Section 4 contains a particularly simple formula for ind(τAB – λ) (in terms of spectral and algebraic invariants of A and B) for the case when τAB – λ is Fredholm (Theorem 4.2). This result is used to prove that (τBA) = –ind(τAB) (Corollary 4.3). We also prove that when A and B are bi-quasi-triangular, then the semi-Fredholm domain of τAB contains no points corresponding to nonzero indices.

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GENERAL FRAMEWORK FOR CONSISTENT SAMPLING IN HILBERT SPACES

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691305000981 ◽

2005 ◽

Vol 03 (04) ◽

pp. 497-509 ◽

Cited By ~ 16

Author(s):

YONINA C. ELDAR ◽

TOBIAS WERTHER

Keyword(s):

Hilbert Spaces ◽

General Framework ◽

Inner Product ◽

Riesz Bases ◽

Oblique Projection ◽

Dual Frames ◽

Infinite Dimensional ◽

Reconstruction Scheme ◽

Linear Reconstruction ◽

Oblique Dual

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinite-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.

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