The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions. Alice and Bob Meet Banach is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information researchers who want to learn AGA or apply its tools; at mathematicians interested in learning QIT, or at least the part of QIT that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. Moreover, this user-friendly book contains numerous tables and explicit estimates, with reasonable constants when possible, which make it a useful reference even for established mathematicians generally familiar with the subject.
The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions. Alice and Bob Meet Banach is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information researchers who want to learn AGA or apply its tools; at mathematicians interested in learning QIT, or at least the part of QIT that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. Moreover, this user-friendly book contains numerous tables and explicit estimates, with reasonable constants when possible, which make it a useful reference even for established mathematicians generally familiar with the subject.
A geometric framework for quantum mechanics arose during the mid 1970s when authors such as Cantoni explored the notion of generalized transition probabilities, and Kibble promoted the idea that the space of pure quantum states provides a natural quantum mechanical analogue for classical phase space. This central idea can be seen easily since the projection of Schrödinger's equation from a Hilbert space into the space of pure spaces is a set of Hamilton's equations. Over the intervening years considerable work has been carried out by a variety of authors and a mature description of quantum mechanics in geometric terms has emerged with many applications. This current offering would seem ideally placed to review the last thirty years of progress and relate this to the most recent work in quantum entanglement. Bengtsson and Zyczkowski's beautifully illustrated volume, Geometry of Quantum States (referred to as GQS from now on) attempts to cover considerable ground in its 466 pages. Its topics range from colour theory in Chapter 1 to quantum entanglement in Chapter 15—to say that this is a whirlwind tour is, perhaps, no understatement. The use of the work 'introduction' in the subtitle of GQS , might suggest to the reader that this work be viewed as a textbook and I think that this interpretation would be incorrect. The authors have chosen to present a survey of different topics with the specific aim to introduce entanglement in geometric terms—the book is not intended as a pedagogical introduction to the geometric approach to quantum mechanics. Each of the fifteen chapters is a short, and mostly self-contained, essay on a particular aspect or application of geometry in the context of quantum mechanics with entanglement being addressed specifically in the final chapter. The chapters fall into three classifications: those concerned with the mathematical background, those which discuss quantum theory and the foundational aspects of the geometric framework, and applications of the geometric approach. The first four chapters contain the standard mathematics required to understand the rest of the material presented: specific areas in colour theory, set theory, probability theory, differential geometry and projective geometry are all covered with an eye to the material that follows. Chapter 5 starts the first real discussion of quantum theory in GQS and serves as an elegant, succinct introduction to the geometry which underlies quantum theory. This may be the most worthwhile chapter for the casual reader who wants to understand the key ideas in this field. Chapter 6 builds on the discussion in Chapter 5, introducing a group theoretic approach to understand coherent states and Chapter 7 describes a geometric tool in the form of an approach to complex projective geometry called 'the stellar representation'. Chapter 8 returns to a more purely quantum mechanical discussion as the authors turn to study the space of density matrices. This chapter completes the discussion which started in Chapter 5. Chapter 9 begins the part of the book concerned with applications of the geometric approach. From this point on the book aims, specifically, to prepare the reader for the material in Chapter 15 beginning with a discussion on the purification of mixed quantum states. In the succeeding chapters a definite choice has been made to present a geometric approach to certain quantum information problems. For example, Chapter 10 contains an extremely well formulated discussion of measurement and positive operator-valued measures with several well illustrated examples and Chapter 11 reopens the discussion of density matrices. Entropy and majorization are again revisited in Chapter 12 in much greater detail than in previous chapters. Chapters 13 and 14 concern themselves with a discussion of various metrics and their relation to the problem of distinguishing between probability distributions and their suitability as probability measures.
Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.
Hilbert spaces, introduced in the early twentieth century as infinite-dimensional generalisations of Euclidean space, provide the rigorous mathematical framework for quantum mechanics and numerous applied disciplines. This paper offers a comprehensive examination of their historical development, axiomatic structure, and multifaceted applications, with particular emphasis on their foundational role in quantum mechanics. We then connect these structures to quantum information geometry, where the (projective) Hilbert space is endowed with natural Riemannian metrics—most notably the Fubini–Study metric for pure states and the Bures metric for mixed states—that quantify statistical distinguishability, parameter estimation precision, and geometric phases. By exploring classical information geometry as a precursor and highlighting quantum extensions, we illustrate how Hilbert-space geometry reveals deep insights into quantum phase transitions, metrology, and quantum technologies. Nuances such as separability, rigged Hilbert spaces for continuous spectra, and infinite-dimensional subtleties are addressed, alongside implications for machine learning via reproducing kernel Hilbert spaces (RKHS) and beyond. The analysis underscores the enduring interplay between pure mathematics and quantum physics. Mamun, SM (2026) …………………………………..1 Hilbert Spaces: Foundations, Applications, and Geometric Connections to Quantum Information Theory (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132277960/download_file?s=portfolio
Introduction The aim of information geometry is to introduce a suitable geometrical structure on families of probability distributions or quantum states. For parametrised statistical models, such structure is based on two fundamental notions: the Fisher information and the exponential family with its dual mixed parametrisation, see for example (Amari 1985, Amari and Nagaoka 2000). For the non-parametric situation, the solution was given by Pistone and Sempi (Pistone and Sempi 1995, Pistone and Rogantin 1999), who introduced a Banach manifold structure on the set P of probability distributions, equivalent to a given one. For each µ ∈ ρ, the authors considered the non-parametric exponential family at µ. As it turned out, this provides a C ∞ -atlas on ρ, with the exponential Orlicz spaces Lφ(µ) as the underlying Banach spaces, here φ is the Young function of the form φ(x) = cosh(x) — 1. The present contribution deals with the case of quantum states: we want to introduce a similar manifold structure on the set of faithful normal states of a von Neumann algebra M. Since there is no suitable definition of a non-commutative Orlicz space with respect to a state φ, it is not clear how to choose the Banach space for the manifold. Of course, there is a natural Banach space structure, inherited from the predualM. But, as it was already pointed out in (Streater 2004), this structure is not suitable to define the geometry of states: for example, any neighbourhood of a state φ contains states such that the relative entropy with respect to φ is infinite.
This paper explores the potential of non-Euclidean geometry to provide a novel framework for understanding and modeling quantum information. Traditional approaches often treat quantum information as a purely mathematical construct, limiting its applicability to purely computational contexts. We propose a radical shift, formalizing quantum information as a geometric object within a non-Euclidean space, leveraging the properties of geometry to reveal deeper connections to the physical world. The core mechanism centers around extending the concept of quantum information beyond classical representations, allowing for the visualization and manipulation of entanglement and other quantum phenomena through a geometric lens. This paper will detail the conceptual groundwork, mathematical framework, and potential implications of this approach, emphasizing its capacity to bridge the gap between abstract mathematics and the complexities of quantum mechanics. Chapter 1: Foundations – Geometry and Information The fundamental concept underpinning this research lies in the recognition that geometry, particularly non-Euclidean geometry, possesses inherent properties that resonate with the nature of quantum information. Classical geometry, with its inherent Euclidean assumptions, struggles to adequately capture the counterintuitive aspects of quantum mechanics. The inherent non-Euclidean nature of space, where distances are not absolute but relative, mirrors the fundamental fuzziness and probabilistic nature of quantum states. Furthermore, the distribution of information, a cornerstone of quantum mechanics, can be visualized as a complex geometric pattern, and non-Euclidean geometry offers a powerful tool to analyze and manipulate this pattern. The challenge, therefore, is to move beyond a purely mathematical abstraction to a geometric understanding of information itself. Early attempts to model quantum information often relied on representations like Hilbert spaces, which, while powerful, represent information as a vector space. However, this representation lacks the inherent geometric richness that is crucial for understanding entanglement and other quantum phenomena. We propose a shift towards representing quantum information as points within a non-Euclidean space, where the distance and orientation of these points encode the quantum state. This shift is predicated on the observation that geometric properties, such as curvature and connectivity, are intimately linked to quantum entanglement and the preservation of information. The core idea is to formulate a geometric model where entanglement is represented not as a correlation between particles but as a distortion of the geometric space itself, influenced by the quantum state. This distortion, driven by the inherent properties of the non-Euclidean space, provides a richer and potentially more intuitive understanding of quantum phenomena. The choice of non-Euclidean geometry is not arbitrary; it is intended to be a 'natural' choice, reflecting the underlying structure of quantum information. Chapter 2: Non-Euclidean Geometry – A Conceptual Framework We introduce a specific type of non-Euclidean geometry, termed "Hyperbolic Geometry for Quantum Information," that incorporates several key features. Unlike Euclidean geometry, Hyperbolic Geometry allows for the representation of multiple, potentially conflicting, states simultaneously. This is achieved through the concept of "Hyperbolic Density," which quantifies the probability of finding a quantum state within a given region of the space. The density of a Hyperbolic Density is influenced by the quantum state, creating a dynamic and evolving geometric representation. The mathematical framework of Hyperbolic Geometry is built around the concept of "Geometric Curvature." This curvature is not simply a measure of distance but rather a representation of the quantum state's inherent orientation. Specifically, we define a curvature function that is sensitive to the quantum state. This curvature is not constant; it adapts to the quantum state, creating a continuously evolving geometric representation. This adaptation is crucial for capturing the inherent randomness and non-determinism of quantum mechanics. Furthermore, we introduce the concept of "Geometric Connectivity," which describes the relationships between points within the Hyperbolic Geometry. These connections are not simply spatial relationships but are defined by the quantum state's inherent properties. The strength and type of connectivity represent the degree to which the quantum state is entangled with the geometric structure. This connectivity is dynamically modified by the curvature function, creating a complex and interconnected geometric pattern. The core mathematical tool for manipulating these geometries is the "Geometric Transformation." This transformation allows us to 'morph' geometric forms into new forms, representing the evolution of quantum information. The transformation is based on a specific set of rules which are intrinsically linked to the nature of entanglement. These rules are carefully designed to preserve the fundamental properties of quantum information.