A Study on the Geometry of Pairs of Positive Linear Forms, Algebraic Transition Probability and Geometrical Phase over Non-Commutative Operator Algebras (II)
The results of the first part [1] will be used to discuss and to investigate some extensions of geometrical notions, which recently have been found to be of interest in Mathematical Physics in context of the problems of the so-called geometrical phase. The concepts of the global phase, the phase group and holonomy group of a normal state of a vN -algebra will be introduced and discussed.
The results of the first part [1] will be used to discuss and to investigate some extensions of geometrical notions, which recently have been found to be of interest in Mathematical Physics in context of the problems of the so-called geometrical phase. The concepts of the global phase, the phase group and holonomy group of a normal state of a vN -algebra will be introduced and discussed.
Some aspects in the geometry of pairs of positive linear forms on unital C* -algebras are considered. Especially, the geometrical relations among the vector representatives of the forms of such a pair within a representation, where both forms can be realized as vectors simultaneously, are studied and discussed in detail. The results obtained in this part extend early results of H. Araki and are intimately related to such functors like the Bures distance and the algebraic transition probability considered by A. Uhlmann and others. The results will be used to discuss and to investigate sonic extensions of geometrical concepts, which have been found to be of interest recently in Mathematical Physics in context of the problems of the so-called geometrical phase.
Geometric phases form a rich subject in mathematical physics, lying at the intersection of differential geometry, functional analysis, and topology. This combination of ideas can make the subject difficult to approach for advanced undergraduate students, even though it is very rewarding. The aim of this thesis is to give a self-contained introduction to the mathematical framework behind geometric phases. Assuming only basic differential geometry, we develop the theory of vector bundles, connections, curvature, and holonomy, and use it to describe the Berry and Wilczek–Zee phases as holonomies of natural connections on complex vector bundles. We also discuss the Aharonov–Bohm effect as an example of a flat connection with globally nontrivial holonomy.
This thesis consists of several studies performed over different few-dof quantum systems exposed to the effect of an uncontrolled environment. The primary focus of the work is to explore the relation between decoherence and environmentally-induced dissipative effects, and the concept known as geometric phases. The first mention of such an object in the context of quantum mechanics goes back to the seminal work by Berry. He demonstrated that the phase acquired by an eigenstate of a time-dependent Hamiltonian in an adiabatic cycle consists of two distinct contributions: one termed 'geometric' and the other known as the dynamical phase. Since Berry's work, the notion of geometric phase has been extended far beyond the original context, encompassing definitions applicable to arbitrary unitary evolutions. These geometric phases naturally arise in the geometric description of Hilbert space, where they manifest as holonomies and possess significance in the fundamental understanding of quantum mechanics and its mathematical framework, and in explaining various physical phenomena, including the Fractional Hall Effect. Moreover, from a modern perspective, geometric phases hold promise for practical applications, such as constructing geometric gates for quantum information processing and storage. However, in practice, a pure state of a quantum system is an idealized concept, and every experimental or real-world implementation must account for the presence of an environment that interacts with the observed system. This interaction necessitates a description in terms of mixed states and non-unitary evolutions. The definition of a geometric phase applicable in such scenarios remains an open problem, giving rise to multiple proposed solutions. Consequently, characterizing these geometric phases encompase motivations from fundamental aspects of quantum mechanics to technological applications.
Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics. Aspects of the representation theory of C*-algebras will be motivated and illustrated in physical terms. Particular emphasis will be given to explicit examples from the theory of quantum phase transitions, where concepts coming from strands as diverse as quantum information theory, algebraic quantum physics and statistical mechanics agreeably converge, providing a more complete picture of the physical phenomena involved.
Abstract. The notion of partial fidelities as invented recently by A.Uhlmann for pairs of finite dimensional density matrices will be extended to the vNalgebraic context and is considered and thoroughly discussed in detail from a mathematical point of view. Especially, in the case of semifinite vN-algebras formulae and estimates for the partial fidelity between the functionals of a dense cone of inner derived normal positive linear forms are obtained. Also, some generalities on the notion of fidelity in quantum physics are collected in an appendix, and another system of mathematical axioms for fidelity over density operators, which is based on the concept of relative majorization and which is intimately related to complete positivity, is proposed. Basic notions and settings. In a vN-algebra M over a Hilbert space H call nonzero x ∈ M+ locally invertible if the linear operator y = x | s(x)H is invertible within s(x)Ms(x), where s(x) is the support of x. Let s(x) ⊥ = 1 − s(x), with the unit operator 1. The local inverse x −1 ∈ M+ of the locally invertible x then is defined as the unique element of M obeying x −1 | s(x)H = y −1 and x −1 | s(x) ⊥ H = 0.
Introduction.The theory of information, created by Shannon [23], is developed by Feinstein, Kullback, MacMillan, Wiener and other American statisticians (e. g., cf.[10]), and also advanced into the ergodic theory by Gelfand, Khinchin, Kolmogorov, Yaglom and other Russian probabilists (e. g., cf.[8]).Through recent years, the theory is regarded as a new chapter in the theory of probability.Recently, Segal [22] gave a mathematical formulation of the entropy of state of a von Neumann algebra, which contains both the cases for the theory of information and the theory of quantum statistics.Segal's theorem was reformulated in operator algebraic form by Nakamura and Umegaki [16] and independently by Davis [3].Since the summer in 1954, Nakamura and Umegaki have investigated the concept of the conditional expectation in von Neumann algebra as a noncommutative extension of probability theory (cf.for example [13~18] and [25 ~28]), and in the most recent paper [18] it was applied to the theory of measurements of quantum statistics which is regarded as a non-commutative case of the theory of entropy and information.Furthermore, it may be very intersesting to develope the theory of information under functional-analysistic and operator-theoretic methods.From these points of views, we shall discuss the measure of information of integrable operators or of normal states of a von Neumann algebra.Davis [3] has independently studied on the almost same theme with Nakamura-Umegaki [16] and [18], in which he developed the theory of entropy and he simplified the proof of the theorem relative to the operatorentropy.Now, we shall give the basic notations and describe the fundamental concepts in a von Neumann algebra which will be used throughout the present paper.Let A be a von Neumann algebra, that is, A is a weakly closed self-adjoint algebra of bounded operators acting over a complex Hubert space H, which contains the identity operator 7. A linear functional p of A is said to be positive if p(aa*) ^ 0 for every a ε A. Such p is said to be state if p(I) = 1, to be normal in the terminology of Dixmier [4] if p(a a )tp(a) for a a ^a, and to be trace if p(ab) = p(ba) for every pair α, b ε A. The normality of state is equivalent to the complete additivity: Σ p(p a ) = p(Σ Pa) for any disjoint family of projections {p a }aA (cf.Dixmier [4]).