Geometrical methods in quantum information are very promising for both providing technical tools and intuition into difficult control or optimization problems. Moreover, they are of fundamental importance in connecting pure geometrical theories, like GR, to quantum mechanics, like in the AdS/CFT correspondence. In this paper, we first make a survey of the most important settings in which geometrical methods have proven useful to quantum information theory. Then we lay down a geometric theory of entanglement by a principle of action, discussing a simple example with two qubits and consequences for a quantum theory of space-time.
Geometrical methods in quantum information are very promising for both providing technical tools and intuition into difficult control or optimization problems. Moreover, they are of fundamental importance in connecting pure geometrical theories, like GR, to quantum mechanics, like in the AdS/CFT correspondence. In this paper, we first make a survey of the most important settings in which geometrical methods have proven useful to quantum information theory. Then we lay down a geometric theory of entanglement by a principle of action, discussing a simple example with two qubits and consequences for a quantum theory of space-time.
The counter-intuitive features of Quantum Mechanics make it possible to solve problems and perform tasks that are beyond the abilities of classical computers and classical communication devices. The area of quantum information processing studies how representing information by quantum states can help achieving such improvements.In this research, we use basic notions of quantum information (mainly entanglement, Bloch sphere, and geometrical distances between quantum states) for analyzing relations of quantum states to each other and quantum cryptographic protocols.Entanglement is an important feature of quantum states. Intuitively (and, partly, inaccurately), entanglement represents quantum (non-classical) correlations between several different quantum systems. Entanglement is one of the most important quantum phenomena, and it has many uses in quantum information, quantum communication, and quantum computing.Some quantum states can be geometrically represented by the Bloch sphere: the unit sphere in the three-dimensional Euclidean space. The "standard" quantum states, to which the laws of Quantum Mechanics directly apply, are called pure states. Other states are the mixed states: probability distributions ("mixtures") of several pure states. The points on the Bloch sphere are the pure states, and those inside the Bloch sphere are the mixed states. This geometrical representation is useful and intuitive for many purposes.We provide a geometrical analysis of entanglement for all the quantum mixed states of rank 2 (all the mixtures of exactly two pure states): for any such state (in any dimension), we define a generalized Bloch sphere by using the two pure states, and we analyze this state and its neighbor states inside this Bloch sphere. We look at the set of non-entangled states ("separable states") in the Bloch sphere and characterize it into exactly five possible classes. We give examples for each class and prove that there are no other classes. In addition, we suggest possible definitions of "entanglement measures" by using the "trace distance" from the nearest separable state.Many types of distances between quantum states can be defined. One of the most useful distances is the trace distance, which bounds the "distinguishability" between the states. The trace distance is very useful in quantum information and quantum cryptography, and it also has a simple geometrical interpretation: it is half of the Euclidean distance between the states in the Bloch sphere.Quantum key distribution (QKD) protocols make it possible for two participators to achieve the classically-impossible task of generating a secret random shared key even if their adversary is computationally unlimited. Several important QKD protocols, including the first protocol of Bennett and Brassard (BB84), have their unconditional security proved against adversaries performing the most general attacks in a theoretical (idealized) setting. We discuss a slightly different protocol, named "BB84-INFO-z", and prove it secure against a broad class of attacks (the collective attacks). Moreover, we make use of the "trace distance" for making our security proof more "composable" than similar security proofs for BB84: namely, for making a step towards proving that the secret key remains secret even when the two participators actually use it for cryptographic purposes.
Quantum Information Geometric Field Theory (QIG-GFT) presents a novel approach to quantum information processing by leveraging the principles of quantum entanglement and superposition to simulate and manipulate geometric spaces. This theory aims to explore the unique geometric properties of quantum computation and quantum information storage through the integration of these quantum phenomena. The core mechanism centers on designing a quantum information processing framework that utilizes quantum entanglement and superposition to achieve geometric simulations, offering a fundamentally different perspective on information handling compared to traditional geometric field theories. This research investigates the potential of this framework to provide a more efficient and intuitive method for addressing complex quantum algorithms and data structures.
This thesis addresses a number of problems within the emerging field of quantum information science. Quantum information science can be said to encompass the more-established disciplines of quantum computation and quantum information, as well as rather more recent attempts to apply concepts, tools and techniques from these disciplines to gain greater understanding of quantum systems in general. The role of entanglement — non-classical correlation — has been of particular interest to date. Part I contributes to this later goal. In particular, we establish a connection between the energy of a many-body quantum system and the idea of an entanglement witness from the theory of mixed-state entanglement. This connection allows mathematical results about entanglement witnesses to be translated into physical results about many-body quantum systems, specifically energy and temperature thresholds for entanglement. For the case of two qubits we are able to establish fairly detailed results about the behaviour of entanglement with temperature. We also study entanglement in systems of indistinguishable particles, where even the question of which quantum states should be regarded as entangled has been the subject of much controversy. We aim to clarify this issue by applying Wiseman and Vaccaro’s notion of entanglement of particles to a number of wellunderstood model systems. We discuss the advantages of the entanglement of particles approach compared with other methods in common use. Finally, we study the operational meaning of superselection rules in quantum physics, in particular the connection to the existence or not of an appropriate reference frame. We propose an experiment that aims to exhibit a coherent superposition of an atom and a molecule, apparently in violation of the commonly-accepted particle-number superselection rule. This result sheds light on the entanglement of particles approach to entanglement of indistinguishable particles. Part II returns to a fundamental question at the heart of quantum computation and quantum information, namely: how many quantum gates are required to perform a particular quantum computation? In other words, how efficiently can a quantum computer solve a particular computational problem? We establish a connection between this question and the field of Riemannian geometry. Intuitively, optimal quantum circuits correspond to “free-falling” along the shortest path between two points in a curved space. This opens up the possibility of using Riemannian geometry to study quantum computation, a possibility that was previously unknown. We provide explicit calculations of all the basic geometric quantities associated with the space, and give some preliminary results of applying geometric ideas to quantum computing. Finally, we explore more generally the connection between optimal control and quantum circuit complexity, of which the Riemannian metric described above can be viewed as a special case.
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.
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.
Many of the tools and ideas of quantum information theory have been employed to shed light on the holographic principle.The entanglement entropy in the AdS/CFT correspondence tells us that information of the bulk geometries is encoded in the boundary correlations.This thesis is aimed to deepen our understanding of this direction by studying various correlation measures in holography.We first derive a generalization of the holographic entanglement of purification conjecture for multipartite states.It turns out that this proposal is consistently supported in arbitrary n-partite states.We next study the entanglement of purification in quantum many body systems by numerical methods.That reveals a remarkable non-monotonicity of the entanglement of purification, attributed to the different sensitivities to quantum and classical correlations.We finally discuss the holographic duals of two classes of correlation measures: optimized correlation measures and axiomatic entanglement measures.The results suggest that classical correlation also plays an important role in encoding the geometrical information.i