Operator algebra

About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.

Theory of Quantum Error Correction for General Noise

Abstract: Quantum error correction protects quantum information against environmental noise. When using qubits, a measure of quality of a code is the maximum number of errors that it is able to correct. We show that a suitable notion of ``number of errors'' e makes sense for any system in the presence of arbitrary environmental interactions. In fact, the notion is directly related to the lowest order in time with which uncorrectable errors are introduced, and this in turn is derived from a grading of the algebra generated by the interaction operators. As a result, e-error-correcting codes are effective at protecting quantum information without requiring the usual assumptions of independence and lack of correlation. We prove the existence of large codes for both quantum and classical information. By viewing error-correcting codes as subsystems, we relate codes to irreducible representations of certain operator algebras and show that noiseless subsystems are infinite-distance error-correcting codes. An explicit example involving collective interactions is discussed.

Isometries of Operator Algebras

Abstract: Well known results of Banach [1]2 and M. H. Stone [8] determine all linear isometric maps of one C(X) onto another (where 'C(X)' denotes, throughout this paper, the set of all real-valued, continuous functions on the compact Hausdorff space X). Such isometries are the maps induced by homeomorphisms of the spaces involved followed by possible changes of sign in the function values on the various closed and open sets. An internal characterization of these isometries would classify them as an algebra isomorphism of the C(X)'s followed by a real unitary multiplication, i.e., multiplication by a real continuous function whose absolute value is 1. The situation in the case of the ring of complex continuous functions (which we denote by 'C'(X)' throughout) is exactly the same; the real unitary multiplication being replaced, of course, by a complex unitary multiplication. It is the purpose of this paper to present the non-commutative extension of the results stated above. A comment as to why this noncommutative extension takes form in a statement about algebras of operators on a Hilbert space seems to be in order. The work of Gelfand-Neumark [2T has as a very particular consequence the fact that each C'(X) is faithfully representable as a self-adjoint, uniformly closed algebra of operators (C*algebra) on a Hilbert space. The representing algebra of operators is, of course, commutative. A statement about the norm and algebraic structure of C' (X) finds then its natural non-commutative extension in the corresponding statement about not necessarily commutative C*algebras. A cursory examination shows that one cannot hope for a word for word transference of the C'(X) result to the non-commutative situation. An isometry between operator algebras is as likely to be an anti-isomorphism as an isomorphism. The direct sum of two C* algebras, which is again a C* algebra, by [2], with an automorphism in one component and an anti-automorphism in the other shows that isomorphisms and anti-isomorphisms together do not encompass all isometries. It is slightly surprising, in view of these facts, that any orderly classification of the isometries of a C* algebra is at all possible. It turns out, in fact, that all isometric maps are composites of a unitary multiplication and a map preserving the C*or quantum mechanical structure (see Segal [7])of the operator algebra in question. More specifically, such maps are linear isomorphisms which commute with the * operation and are multiplicative on powers, composed with a multiplication by a unitary operator in the algebra.

Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. i. mapping theorems and ordering of functions of noncommuting operators.

Abstract: A new calculus for functions of noncommuting operators is developed, based on the notion of mapping of functions of operators onto $c$-number functions. The class of linear mappings, each member of which is characterized by an entire analytic function of two complex variables, is studied in detail. Closed-form solutions for such mappings and for the inverse mappings are obtained and various properties of these mappings are studied. It is shown that the most commonly occurring rules of association between operators and $c$-numbers (the Weyl, the normal, the antinormal, the standard, and the antistandard rules) belong to this class and are, in fact, the simplest ones in a clearly defined sense. It is shown further that the problem of expressing an operator in an ordered form according to some prescribed rule is equivalent to an appropriate mapping of the operator on a $c$-number space. The theory provides a systematic technique for the solution of numerous quantum-mechanical problems that were treated in the past by ad hoc methods, and it furnishes a new approach to many others. This is illustrated by a number of examples relating to mappings and ordering of operators.

Superselection sectors with braid group statistics and exchange algebras. I: General theory

Abstract: The theory of superselection sectors is generalized to situations in which normal statistics has to be replaced by braid group statistics. The essential role of the positive Markov trace of algebraic quantum field theory for this analysis is explained, and the relation to exchange algebras is established.