scholarly journals On the equivalence of separability and extendability of quantum states

2017 ◽  
Vol 29 (04) ◽  
pp. 1750012 ◽  
Author(s):  
B. V. Rajarama Bhat ◽  
K. R. Parthasarathy ◽  
Ritabrata Sengupta
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Gaussian State ◽  
Bounded Operators ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Separability Property ◽  
Quantum De Finetti Theorem ◽  
Necessary And Sufficient ◽  
Symmetric States

Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.

FROM QUANTUM FIELDS TO LOCAL VON NEUMANN ALGEBRAS

1992 ◽  
Vol 04 (spec01) ◽  
pp. 15-47 ◽  
Author(s):  
H.J. BORCHERS ◽  
JAKOB YNGVASON
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Energy Bounds ◽  
Local Algebras ◽  
Quantized Fields ◽  
Necessary And Sufficient

The subject of the paper is an old problem of the general theory of quantized fields: When can the unbounded operators of a Wightman field theory be associated with local algebras of bounded operators in the sense of Haag? The paper reviews and extends previous work on this question, stressing its connections with a noncommutive generalization of the classical Hamburger moment problem. Necessary and sufficient conditions for the existence of a local net of von Neumann algebras corresponding to a given Wightman field are formulated in terms of strengthened versions of the usual positivity property of Wightman functionals. The possibility that the local net has to be defined in an enlarged Hilbert space cannot be ruled out in general. Under additional hypotheses, e.g., if the field operators obey certain energy bounds, such an extension of the Hilbert space is not necessary, however. In these cases a fairly simple condition for the existence of a local net can be given involving the concept of “central positivity” introduced by Powers. The analysis presented here applies to translationally covariant fields with an arbitrary number of components, whereas Lorentz covariance is not needed. The paper contains also a brief discussion of an approach to noncommutative moment problems due to Dubois-Violette, and concludes with some remarks on modular theory for algebras of unbounded operators.

Unitary equivalence and reductibility or invertibly weighted shifts

1971 ◽  
Vol 5 (2) ◽  
pp. 157-173 ◽  
Author(s):  
Alan Lambert
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Sequence ◽  
Complex Hilbert Space ◽  
Weighted Shifts ◽  
Infinite Dimensional ◽  
Reducing Subspaces ◽  
Positive Weights ◽  
Completely Reducible ◽  
Necessary And Sufficient

Let H be a complex Hilbert space and let {A1, A2, …} be a uniformly bounded sequence of invertible operators on H. The operator S on l2(H) = H ⊕ H ⊕ … given by S〈x0, x1, …〉 = 〈0, A1x0, A2x1, …〉 is called the invertibly veighted shift on l2(H) with weight sequence {An }. A matricial description of the commutant of S is established and it is shown that S is unitarily equivalent to an invertibly weighted shift with positive weights. After establishing criteria for the reducibility of S the following result is proved: Let {B1, B2, …} be any sequence of operators on an infinite dimensional Hilbert space K. Then there is an operator T on K such that the lattice of reducing subspaces of T is isomorphic to the corresponding lattice of the W* algebra generated by {B1, B2, …}. Necessary and sufficient conditions are given for S to be completely reducible to scalar weighted shifts.

The Approach to Equilibrium of a Class of Quantum Dynamical Semigroups

1998 ◽  
Vol 01 (04) ◽  
pp. 561-572 ◽  
Author(s):  
Franco Fagnola ◽  
Rolando Rebolledo
Keyword(s):  
Hilbert Space ◽  
Asymptotic Behavior ◽  
Conditional Expectation ◽  
Sufficient Conditions ◽  
Approach To Equilibrium ◽  
Bounded Operators ◽  
Quantum Dynamical Semigroup ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Necessary And Sufficient

This paper deals with the asymptotic behavior of a quantum dynamical semigroup [Formula: see text] acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a generator which can be written in a well-known form named after Lindblad and Davies. If the semigroup has a faithful normal stationary state ρ, necessary and sufficient conditions are derived for the w*-convergence of [Formula: see text] to [Formula: see text], where [Formula: see text] is the conditional expectation of an element X onto the subalgebra of fixed points. Our main results are expressed in terms of the Lindblad–Davies generator .

On the Operator Identity ∑ AkXBk ≡ 0

1978 ◽  
Vol 31 (4) ◽  
pp. 845-857 ◽  
Author(s):  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Operator Identity ◽  
Image Position ◽  
Space Case ◽  
Necessary And Sufficient ◽  
Induced Mapping ◽  
Compact Map

Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by(1)We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the formFor example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

TO A TENSOR ANALYSIS OF THE SOLVABILITY OF THE PROBLEM OF REALIZATION OF A SECOND-ORDER BILINEAR SYSTEM WITH DELAY

2021 ◽  
Vol 2 ◽  
pp. 79-92
Author(s):  
Anatoly Lakeyev ◽  
◽  
Vyacheslav Rusanov ◽  
Andrey Banshchikov ◽  
◽  
...  
Keyword(s):  
Hilbert Space ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Bilinear System ◽  
Second Order ◽  
Infinite Dimensional ◽  
Type Property ◽  
Machine Interface ◽  
Linear Differential ◽  
Necessary And Sufficient

The analytical conditions (necessary and sufficient) are defined for the solvability of the problem of differential realization of a continuous beam of controlled trajectory curves in the class of bilinear nonautonomous ordinary differential equations (with delay and without it) of the second order in a real separable Hilbert space. The problem under consideration belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolution equations in an infinite-dimensional Hilbert space. The meta-language of this theory is the constructions of tensor products of Hilbert spaces, the structures of lattices with ortho-complementation, and the functional apparatus of the nonlinear Rayleigh-Ritz operator. It is shown that in the case of a finite bundle of trajectories, the presence of a sublinearity-type property of this operator allows us to obtain sufficient conditions for the existence of such realizations. Along the way, the topological-metric conditions for the continuity of the projectivization of the nonlinear Rayleigh-Ritz functional operator with the calculation of the fundamental group of its image are justified. The results obtained provide the motivation for the development of a qualitative theory of nonlinear structural identification of higher-order multi-linear differential models (e.g. for processes, induced by the «brain–machine» interface-platform of the type of Neuralink).

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

scholarly journals Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

scholarly journals On an infinite dimensional linear-quadratic problem with fixed endpoints: The continuity question

2014 ◽  
Vol 24 (4) ◽  
pp. 723-733
Author(s):  
K.Maciej Przyłuski
Keyword(s):  
Minimum Energy ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Linear Quadratic Control ◽  
Linear Quadratic ◽  
Minimum Norm Solution ◽  
Quadratic Problem ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Quadratic Control Problems

Abstract In a Hilbert space setting, necessary and sufficient conditions for the minimum norm solution u to the equation Su = Rz to be continuously dependent on z are given. These conditions are used to study the continuity of minimum energy and linear-quadratic control problems for infinite dimensional linear systems with fixed endpoints.

scholarly journals When do Projections Commute?

1980 ◽  
Vol 35 (4) ◽  
pp. 437-441 ◽  
Author(s):  
W. Rehder
Keyword(s):  
Hilbert Space ◽  
Conditional Probability ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantum Mechanical ◽  
Probability Operator ◽  
Quantum Mechanical Systems ◽  
Necessary And Sufficient ◽  
New Criteria

Abstract Necessary and sufficient conditions for commutativity of two projections in Hilbert space are given through properties of so-called conditional connectives which are derived from the conditional probability operator PQP. This approach unifies most of the known proofs, provides a few new criteria, and permits certain suggestive interpretations for compound properties of quantum-mechanical systems.

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