Quantum information theory has become a central area in the two fields of quantum mechanics and information science. The focus of this work is mathematical modelling of quantum information with the use of operator algebras, in a way that accentuates the modelling, and acting or manipulation of observables and quantum states. The fundamental aim is to utilise C*-algebras, von Neumann algebras, and other frameworks in operator algebra to formally describe the behaviour of quantum systems, understand the nature of quantum entanglement, and support quantum error correction. Operator algebra usage provides a firm mathematical foundation that guarantees accuracy and contributes to the incorporation of topological mechanisms and algebraic rules necessary for explaining noncommutative concepts in quantum mechanics. In this paper, we will present some original notions of operator algebra and explain their usefulness when it comes to modeling the key dynamics, features of quantum states, measurement, and evolution. We will use a method that combines algebraic techniques to chart qubit interactions, entropic inequalities, and characterize transformations of level in unitary evolution. Through the use of Hilbert spaces and bounded linear operators, we build models in multi-qubit systems and see what the specifications have to say about quantum computing and quantum communication protocols. The paper provide a comprehensive discussion, including algebraic formulations, simulation outputs, and a critical comparison with traditional Hilbert space methods. It is quite notable that the operator algebras can say more about the non-local correlations and decoherence processes. There are extensive literature reviews, useful modelling tools, and implementation outcomes in the paper when symbolic computation tools have been utilised. This combination gives both theoretical strength and modelling instruments to develop quantum technologies in the future.
Quantum information theory has become a central area in the two fields of quantum mechanics and information science. The focus of this work is mathematical modelling of quantum information with the use of operator algebras, in a way that accentuates the modelling, and acting or manipulation of observables and quantum states. The fundamental aim is to utilise C*-algebras, von Neumann algebras, and other frameworks in operator algebra to formally describe the behaviour of quantum systems, understand the nature of quantum entanglement, and support quantum error correction. Operator algebra usage provides a firm mathematical foundation that guarantees accuracy and contributes to the incorporation of topological mechanisms and algebraic rules necessary for explaining noncommutative concepts in quantum mechanics. In this paper, we will present some original notions of operator algebra and explain their usefulness when it comes to modeling the key dynamics, features of quantum states, measurement, and evolution. We will use a method that combines algebraic techniques to chart qubit interactions, entropic inequalities, and characterize transformations of level in unitary evolution. Through the use of Hilbert spaces and bounded linear operators, we build models in multi-qubit systems and see what the specifications have to say about quantum computing and quantum communication protocols. The paper provide a comprehensive discussion, including algebraic formulations, simulation outputs, and a critical comparison with traditional Hilbert space methods. It is quite notable that the operator algebras can say more about the non-local correlations and decoherence processes. There are extensive literature reviews, useful modelling tools, and implementation outcomes in the paper when symbolic computation tools have been utilised. This combination gives both theoretical strength and modelling instruments to develop quantum technologies in the future.
A central problem in modern mathematical physics is to study interacting systems beyond the reach of perturbation theory. This thesis studies interacting quantum systems using the mathematical theory of operator algebras. In the first part of this thesis, we consider how to construct quantum many-body systems from one-particle systems. This problem has a well-established solution - namely, the second quantization (or the Gaussian functor). However, it seems difficult for the second quantization to go beyond generalized free field constructions without the help of perturbation theory. The second quantization takes real Hilbert spaces as inputs. Using only information from one-particle Hilbert spaces, it is mathematically impossible for the second quantization to produce connected higher-point correlation functions. Moreover, from the perspectives of mathematical quantization and operator algebra theory, it is more natural to have a quantization procedure that takes an algebra along with a functional as the input. Such a pair of mathematical objects contains more information about the underlying one-particle system and includes the Hilbert space of states (via the GNS construction) as part of the data. Inspired by quantum probability theory, we introduce an alternative approach to construct quantum many-body systems. This approach is based on a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Mathematically, Poissonization is a functor from the category of von Neumann algebras with normal semi nite faithful weights to the category of von Neumann algebras with normal faithful states. It is a natural adaptation of the second quantization to the context of von Neumann algebras. This thesis discusses several properties of Poissonization, and uses Poissonization as a tool to construct various toy models of algebraic quantum field theories relevant to high energy physics. This collection of examples aims at demonstrating the versatility of Poissonization. In the second part of this thesis, we study the noisy interactions between an open quantum system and its environment. One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's. In this thesis, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state.
Abstract Quantum information science is a new and rapidly developing discipline that studies the general laws governing the transmission, storage, and processing of information in systems that obey the laws of quantum mechanics. It makes extensive use of matrix theory and operator theory on Hilbert spaces to investigate the capabilities of such systems and to develop principles for their rational design. An important consequence of the development of quantum information science is a substantial clarification of the logical structure and foundations of quantum mechanics and of its relation to reality. These lectures are devoted to the foundations of the mathematical theory of quantum systems: states, observables, communication channels, their entropic and information characteristics, and quantum information-processing protocols. The lectures provide a detailed introduction to the statistical structure of quantum theory and the necessary background from classical information theory; hence, knowledge of the basic mathematical subjects—analysis, linear algebra, and probability theory—is sufficient for understanding the material. In the first part of the course, we use the “standard” statistical model of a quantum system, but consider it in comparison with the classical statistical description. A consistent development of this approach leads to a generalized statistical model of quantum mechanics, in which observables (in general, “unsharp”) are described by probability operator-valued measures, while evolutions (in general, irreversible) are described by completely positive maps. Introductory material on these concepts, which underlie mathematical models of quantum information systems in the presence of noise and errors and have numerous applications in quantum information science, is presented in the second part of the course.
This paper explores the intricate connections between various algebraic structures—Banach algebras, C*-algebras, and von Neumann algebras—and their profound implications for the theory of quantum entanglement and quantum information. We establish a comprehensive framework that bridges the abstract concepts of functional analysis with the concrete demands of quantum mechanics. Beginning with the foundational properties of Banach algebras, we progressively introduce C*-algebras as a natural setting for quantum observables and states, followed by von Neumann algebras, which are essential for describing quantum measurement and infinite-dimensional quantum systems. A central focus is to delineate how quantum entanglement, a cornerstone resource in quantum information science, can be rigorously characterized and quantified within this hierarchical algebraic framework. We integrate recent game-theoretic approaches to quantum information, providing new perspectives on quantum relative entropy, mutual information, and channel capacity. By demonstrating how the rich structure of C*-algebras acts as a pivotal bridge, we illuminate the mathematical coherence underlying the transition from general algebraic concepts to the specialized domain of quantum information processing, offering a unified view of quantum phenomena. The work also delves into the operational justifications for these algebraic descriptions, particularly in the context of infinite quantum systems.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.