scholarly journals ENTANGLEMENT, HAAG-DUALITY AND TYPE PROPERTIES OF INFINITE QUANTUM SPIN CHAINS

2006 ◽  
Vol 18 (09) ◽  
pp. 935-970 ◽  
Author(s):  
M. KEYL ◽  
T. MATSUI ◽  
D. SCHLINGEMANN ◽  
R. F. WERNER
Keyword(s):  
Hilbert Spaces ◽  
Quantum Spin ◽  
Xy Model ◽  
Type I ◽  
Quantum Spin Chains ◽  
Haag Duality ◽  
Von Neumann ◽  
Neumann Type ◽  
Left And Right ◽  
Normal States

We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.

scholarly journals ON HAAG DUALITY FOR PURE STATES OF QUANTUM SPIN CHAINS

2008 ◽  
Vol 20 (06) ◽  
pp. 707-724 ◽  
Author(s):  
M. KEYL ◽  
TAKU MATSUI ◽  
D. SCHLINGEMANN ◽  
R. F. WERNER
Keyword(s):  
Von Neumann Algebra ◽  
Quantum Spin ◽  
Spin Chains ◽  
Type I ◽  
Quantum Spin Chains ◽  
Haag Duality ◽  
Von Neumann ◽  
Infinite Intervals ◽  
Pure States

In this note, we consider quantum spin chains and their translationally invariant pure states. We prove Haag duality for quasilocal observables localized in semi-infinite intervals (-∞ , 0] and [1, ∞) when the von Neumann algebra generated by observables localized in [0, ∞) is non-type I.

scholarly journals ON PRODUCT SYSTEMS ARISING FROM SUM SYSTEMS

2005 ◽  
Vol 08 (01) ◽  
pp. 1-31 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
R. SRINIVASAN
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann Algebra ◽  
Type I ◽  
Product System ◽  
Von Neumann ◽  
Type Iii ◽  
Product Systems ◽  
Weyl Representation ◽  
Uncountable Family ◽  
Type Spaces

B. Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gaussian spaces, measure type spaces and "slightly colored noises", using techniques from probability theory. Here we take a purely functional analytic approach and try to have a better understanding of Tsireleson's construction and his examples. We prove an extension of Shale's theorem connecting symplectic group and Weyl representation. We show that the "Shale map" respects compositions (this settles an old conjecture of K. R. Parthasarathy8). Using this we associate a product system to a sum system. This construction includes the exponential product system of Arveson, as a trivial case, and the type III examples of Tsirelson. By associating a von Neumann algebra to every "elementary set" in [0, 1], in a much simpler and direct way, we arrive at the invariants of the product system introduced by Tsirelson, given in terms of the sum system. Then we introduce a notion of divisibility for a sum system, and prove that the examples of Tsirelson are divisible. It is shown that only type I and type III product systems arise out of divisible sum systems. Finally, we give a sufficient condition for a divisible sum system to give rise to a unitless (type III) product system.

scholarly journals BOUNDEDNESS OF ENTANGLEMENT ENTROPY AND SPLIT PROPERTY OF QUANTUM SPIN CHAINS

2013 ◽  
Vol 25 (09) ◽  
pp. 1350017 ◽  
Author(s):  
TAKU MATSUI
Keyword(s):  
Spectral Gap ◽  
Entanglement Entropy ◽  
Quantum Spin ◽  
Spin Systems ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Quantum Spin Systems ◽  
One Dimensional ◽  
Split Property ◽  
Left And Right

We show that boundedness of entanglement entropy for pure states of bipartite quantum spin systems implies split property of subsystems. As a corollary, in one-dimensional quantum spin chains, we show that the split property with respect to left and right semi-infinite subsystems is valid for the translationally invariant pure ground states with spectral gap.

Solvable nineteen-vertex models and quantum spin chains of spin one

1994 ◽  
Vol 4 (8) ◽  
pp. 1151-1159 ◽  
Author(s):  
Makoto Idzumi ◽  
Tetsuji Tokihiro ◽  
Masao Arai
Keyword(s):  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Vertex Models

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

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