Infinitely entangled states

2003 ◽  
Vol 3 (4) ◽  
pp. 281-306
Author(s):  
M. Keyl ◽  
D. Schlingemann ◽  
R.F. Werner
Keyword(s):  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Entangled States ◽  
Entangled State ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Density Operators ◽  
Fundamental Paper ◽  
Extended Notion

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.

scholarly journals MINIMUM UNCERTAINTY AND ENTANGLEMENT

2013 ◽  
Vol 27 (16) ◽  
pp. 1350068 ◽  
Author(s):  
N. D. HARI DASS ◽  
TABISH QURESHI ◽  
ADITI SHEEL
Keyword(s):  
Hilbert Spaces ◽  
Uncertainty Relation ◽  
Entangled State ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
System A ◽  
Finite Dimensional ◽  
Schmidt Rank ◽  
Heisenberg Uncertainty ◽  
Minimum Uncertainty

We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrödinger–Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there is a class of physically interesting entangled states for which no two noncommuting observables can attain the minimum uncertainty equality.

scholarly journals Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
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.

scholarly journals Multidimensional entanglement transport through single-mode fiber

2020 ◽  
Vol 6 (4) ◽  
pp. eaay0837 ◽  
Author(s):  
Jun Liu ◽  
Isaac Nape ◽  
Qainke Wang ◽  
Adam Vallés ◽  
Jian Wang ◽  
...  
Keyword(s):  
Hilbert Spaces ◽  
Mode Coupling ◽  
Degrees Of Freedom ◽  
Single Mode ◽  
Single Mode Fiber ◽  
Entangled States ◽  
Spatial Mode ◽  
Alternative Approach ◽  
Spatial Modes ◽  
State Tomography

The global quantum network requires the distribution of entangled states over long distances, with substantial advances already demonstrated using polarization. While Hilbert spaces with higher dimensionality, e.g., spatial modes of light, allow higher information capacity per photon, such spatial mode entanglement transport requires custom multimode fiber and is limited by decoherence-induced mode coupling. Here, we circumvent this by transporting multidimensional entangled states down conventional single-mode fiber (SMF). By entangling the spin-orbit degrees of freedom of a biphoton pair, passing the polarization (spin) photon down the SMF while accessing multiple orbital angular momentum (orbital) subspaces with the other, we realize multidimensional entanglement transport. We show high-fidelity hybrid entanglement preservation down 250 m SMF across multiple 2 × 2 dimensions, confirmed by quantum state tomography, Bell violation measures, and a quantum eraser scheme. This work offers an alternative approach to spatial mode entanglement transport that facilitates deployment in legacy networks across conventional fiber.

scholarly journals A SIMPLE PROOF OF THE FUNDAMENTAL THEOREM ABOUT ARVESON SYSTEMS

2006 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
MICHAEL SKEIDE
Keyword(s):  
Hilbert Space ◽  
Simple Proof ◽  
Fundamental Theorem ◽  
Hilbert Module ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Product Systems ◽  
Dimensional Hilbert Space ◽  
The One

With every E0-semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0-semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and allows for a generalization to product systems of Hilbert module (to be published elsewhere).

scholarly journals TOWARDS A DEFINITION OF QUANTUM INTEGRABILITY

2009 ◽  
Vol 06 (01) ◽  
pp. 129-172 ◽  
Author(s):  
JESÚS CLEMENTE-GALLARDO ◽  
GIUSEPPE MARMO
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Complete Integrability ◽  
Infinite Dimensional ◽  
Quantum Integrability ◽  
Classical Systems ◽  
Definition Of ◽  
Geometrical Formulation ◽  
Quantum Framework

We briefly review the most relevant aspects of complete integrability for classical systems and identify those aspects which should be present in a definition of quantum integrability. We show that a naive extension of classical concepts to the quantum framework would not work because all infinite dimensional Hilbert spaces are unitarilly isomorphic and, as a consequence, it would not be easy to define degrees of freedom. We argue that a geometrical formulation of quantum mechanics might provide a way out.

scholarly journals Tomography in Abstract Hilbert Spaces

2006 ◽  
Vol 13 (03) ◽  
pp. 239-253 ◽  
Author(s):  
V. I. Man'ko ◽  
G. Marmo ◽  
A. Simoni ◽  
F. Ventriglia
Keyword(s):  
Hilbert Space ◽  
Quantum State ◽  
Trace Formula ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Rank One

The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.

Strong Extensions vs. Weak Extensions of C*-Algebras

1978 ◽  
Vol 21 (2) ◽  
pp. 143-147
Author(s):  
S. J. Cho
Keyword(s):  
Hilbert Space ◽  
Partial Isometry ◽  
Compact Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Image Position ◽  
The Ideal

Let be a separable complex infinite dimensional Hilbert space, the algebra of bounded operators in the ideal of compact operators, and the quotient map. Throughout this paper A denotes a separable nuclear C*-algebra with unit. An extension of A is a unital *-monomorphism of A into . Two extensions τ1 and τ2 are strongly (weakly) equivalent if there exists a unitary (Fredholm partial isometry) U in such thatfor all a in A.

scholarly journals On the Relation between States and Maps in Infinite Dimensions

2007 ◽  
Vol 14 (04) ◽  
pp. 355-370 ◽  
Author(s):  
Janusz Grabowski ◽  
Marek Kuś ◽  
Giuseppe Marmo
Keyword(s):  
Hilbert Spaces ◽  
Trace Class ◽  
Quantum Systems ◽  
Bounded Operators ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Continuous Map ◽  
Finite Dimensional ◽  
Trace Class Operators ◽  
Product Realization

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

scholarly journals Infinite-dimensional numerical linear algebra: theory and applications

2010 ◽  
Vol 466 (2124) ◽  
pp. 3539-3559 ◽  
Author(s):  
Anders C. Hansen
Keyword(s):  
Linear Algebra ◽  
Hilbert Spaces ◽  
Numerical Linear Algebra ◽  
New Method ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
The Core ◽  
Algebra Theory ◽  
Spectra And Pseudospectra

We present a new method for computing spectra and pseudospectra of bounded operators on separable Hilbert spaces. The core in this theory is a generalization of the pseudospectrum called the n -pseudospectrum.

scholarly journals Products of Positive Operators

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
Maximiliano Contino ◽  
Michael A. Dritschel ◽  
Alejandra Maestripieri ◽  
Stefania Marcantognini
Keyword(s):  
Positive Operator ◽  
Hilbert Spaces ◽  
Spectral Properties ◽  
Compact Operators ◽  
Positive Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Special Classes

AbstractOn finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class $${\mathcal {L}^{+\,2}}$$ L + 2 of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in $${\mathcal {L}^{+\,2}}$$ L + 2 are developed, and membership in $${\mathcal {L}^{+\,2}}$$ L + 2 among special classes, including algebraic and compact operators, is examined.

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