On Extending the Trace as a Linear Functional, II

1980 ◽  
Vol 32 (6) ◽  
pp. 1288-1298
Author(s):  
George A. Elliott
Keyword(s):  
Positive Operator ◽  
Selfadjoint Operator ◽  
Linear Functional ◽  
Orthonormal Basis ◽  
Unitary Operator ◽  
Trace Class ◽  
Diagonal Matrix ◽  
Matrix Elements ◽  
Von Neumann ◽  
The Difference

A positive bounded selfadjoint operator is in the trace class of von Neumann and Schatten ([4]) if the sum of its diagonal matrix elements with respect to some orthonormal basis is finite, and the trace is then defined to be this sum, which is independent of the basis. A bounded selfadjoint but not necessarily positive operator x is in the trace class if in the decomposition x = x+ – x−, with x+ and x− positive and x+x− = 0, both x+ and x−are in the trace class; the trace of x is then defined to be the difference of the finite traces of x+ and x−. The trace defined in this way is a linear functional on the trace class, and is unitarily invariant; if u is a unitary operator, the trace of uxu−1 is the same as the trace of x.

scholarly journals Quantum Orlicz Spaces in Information Geometry

2004 ◽  
Vol 11 (04) ◽  
pp. 359-375 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Quantum Information ◽  
Tangent Space ◽  
Selfadjoint Operator ◽  
Affine Connection ◽  
Orlicz Spaces ◽  
Trace Class ◽  
Information Geometry ◽  
Young Function ◽  
Affine Structure ◽  
Cotangent Space

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

INVESTIGATION COMPARISON ON THE GATE SPEED OF THREE-LEVEL AND TWO-LEVEL QUANTUM NOT GATES WITH ASYMMETRIC POTENTIAL OF rf-SQUID

2007 ◽  
Vol 21 (19) ◽  
pp. 1261-1270 ◽  
Author(s):  
YING-HUA JI ◽  
JU-JU HU ◽  
SHI-HUA CAI
Keyword(s):  
Microwave Pulse ◽  
Diagonal Matrix ◽  
Matrix Elements ◽  
The Asymmetry ◽  
Rf Squid ◽  
Double Well Potential

We investigate the relation between the speed of quantum NOT gate and the asymmetry or detuning of the potential in system of the interaction of a two-level rf-SQUID qubit with a classical microwave pulse. The rf-SQUID is characterized by an asymmetric double well potential that gives rise to diagonal matrix elements. Then in resonance, we compare the gate speeds for three-level and two-level quantum NOT gates. We show that in general, a three-level gate is much faster than the conventional two-level gate.

Broadband EM radiation amplification by means of a monochromatically driven two-level system

2017 ◽  
Vol 31 (04) ◽  
pp. 1750027 ◽  
Author(s):  
Andrey V. Soldatov
Keyword(s):  
Dipole Moment ◽  
Quantum System ◽  
High Frequency ◽  
Laser Field ◽  
Diagonal Matrix ◽  
Matrix Elements ◽  
Level System ◽  
Dipole Moment Operator ◽  
Moment Operator ◽  
Radiation Amplification

It is shown that a two-level quantum system possessing dipole moment operator with permanent non-equal diagonal matrix elements and driven by external semiclassical monochromatic high-frequency electromagnetic (EM) (laser) field can amplify EM radiation waves of much lower frequency.

DIAGONALIZATION MODULO NORM IDEALS AND HAUSDORFF DIMENSIONALITY

2000 ◽  
Vol 11 (08) ◽  
pp. 1057-1078
Author(s):  
JINGBO XIA
Keyword(s):  
Hausdorff Measure ◽  
Metric Spaces ◽  
Adjoint Operator ◽  
Trace Class ◽  
Multiplication Operators ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
One Dimensional ◽  
Compact Metric Spaces ◽  
The One

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal [Formula: see text] not contained in the trace class [Formula: see text], every self-adjoint operator A admits the decomposition A=D+K, where D is a self-adjoint diagonal operator and [Formula: see text]. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if μ is a regular Borel measure on X which has a σ-finite one-dimensional Hausdorff measure, then the family {Mf:f∈ Lip (X)} of multiplication operators on T2(X, μ) can be simultaneously diagonalized modulo any [Formula: see text]. Because the condition [Formula: see text] in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between [Formula: see text] and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p>1.

scholarly journals Convergence Conditions of Mixed States and their von Neumann Entropy in Continuous Quantum Measurements

2014 ◽  
Vol 21 (04) ◽  
pp. 1450010
Author(s):  
Toru Fuda
Keyword(s):  
Quantum Measurements ◽  
Von Neumann Entropy ◽  
Projection Operators ◽  
Mixed States ◽  
Convergence Conditions ◽  
Von Neumann ◽  
Zeno Effect ◽  
The Difference ◽  
Arbitrary Precision ◽  
Continuous Quantum Measurements

By carrying out appropriate continuous quantum measurements with a family of projection operators, a unitary channel can be approximated in an arbitrary precision in the trace norm sense. In particular, the quantum Zeno effect is described as an application. In the case of an infinite dimension, although the von Neumann entropy is not necessarily continuous, the difference of the entropies between the states, as mentioned above, can be made arbitrarily small under some conditions.

On the Unitary Equivalence of Certain Classes of Non-Normal Operators. I

1971 ◽  
Vol 23 (5) ◽  
pp. 849-856 ◽  
Author(s):  
P. K. Tam
Keyword(s):  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Equivalence Problem ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Type I ◽  
Unitary Equivalence ◽  
Bounded Linear ◽  
Von Neumann ◽  
Normal Operators

The following (so-called unitary equivalence) problem is of paramount importance in the theory of operators: given two (bounded linear) operators A1, A2 on a (complex) Hilbert space , determine whether or not they are unitarily equivalent, i.e., whether or not there is a unitary operator U on such that U*A1U = A2. For normal operators this question is completely answered by the classical multiplicity theory [7; 11]. Many authors, in particular, Brown [3], Pearcy [9], Deckard [5], Radjavi [10], and Arveson [1; 2], considered the problem for non-normal operators and have obtained various significant results. However, most of their results (cf. [13]) deal only with operators which are of type I in the following sense [12]: an operator, A, is of type I (respectively, II1, II∞, III) if the von Neumann algebra generated by A is of type I (respectively, II1, II∞, III).

Perturbations of AF-Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1012-1016 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Image Position ◽  
Af Algebras

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

On Products of Symmetries

1966 ◽  
Vol 18 ◽  
pp. 897-900 ◽  
Author(s):  
Peter A. Fillmore
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Type Ii ◽  
Central Projections ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Principal Tool ◽  
The Subject

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebraA(i.e., one with no finite non-zero central projections), then the symmetries may be chosen fromA.A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

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