This work explores how geometry shapes the rules of quantum mechanics. Just as planets move along curved paths in space, quantum states evolve on geometric “phase spaces” with unique shapes. We show how these spaces like the famous Bloch sphere for qubits help us understand quantum dynamics, speed limits, and even entanglement. A key focus is the “geometric phase,” a memory-like effect acquired by quantum systems during cyclic journeys, akin to a hiker gaining elevation on a mountain trail. We solve the “quantum brachistochrone” problem finding the fastest path between quantum states using geometric tools. For two interacting spins, we reveal how entanglement warps the geometry of their shared state space, linking stronger correlations to flatter curvature. Finally, we connect these ideas to quantum computing, showing how geometric phases enable robust logic gates in experiments like nuclear magnetic resonance (NMR). This bridges abstract math to real-world quantum technologies.
This work explores how geometry shapes the rules of quantum mechanics. Just as planets move along curved paths in space, quantum states evolve on geometric “phase spaces” with unique shapes. We show how these spaces like the famous Bloch sphere for qubits help us understand quantum dynamics, speed limits, and even entanglement. A key focus is the “geometric phase,” a memory-like effect acquired by quantum systems during cyclic journeys, akin to a hiker gaining elevation on a mountain trail. We solve the “quantum brachistochrone” problem finding the fastest path between quantum states using geometric tools. For two interacting spins, we reveal how entanglement warps the geometry of their shared state space, linking stronger correlations to flatter curvature. Finally, we connect these ideas to quantum computing, showing how geometric phases enable robust logic gates in experiments like nuclear magnetic resonance (NMR). This bridges abstract math to real-world quantum technologies.
This thesis, explores the quantum entanglement and evolution through both a geometric and dynamical perspective. The first part focuses on classical phase space and its central role in Hamiltonian mechanics, emphasizing the importance of symplectic structures in describing mechanical states. The study highlights the formal analogy between classical phase space and the Hilbert space used in quantum mechanics. The second part is devoted to the geometric description of quantum states through the projective structure of Hilbert space. Emphasis is placed on the geometric interpretation of quantum evolution, particularly via the Fubini-Study metric, associated symplectic structures, and the geometric phase acquired during unitary evolutions. The final two parts are dedicated to the study of spin systems (both two-body and many-body) under different interaction models (XXZ Heisenberg and all-range Ising). Both the dynamical aspects (evolution speed, entanglement, and the quantum brachistochrone problem) and the geometric and topological structures of the corresponding quantum states are analyzed.
We present a unified geometric and dynamical framework for a physical system consisting of n spin-1/2 particles with all-range Ising interaction. Using the Fubini-Study formalism, we derive the metric tensor of the associated quantum state manifold and compute the corresponding Riemann curvature. Our analysis reveals that the system evolves over a smooth, compact, two-dimensional manifold with spherical topology and a dumbbell-like structure shaped by collective spin interactions. We further investigate the influence of the geometry and topology of the resulting state space on the behavior of geometric and topological phases acquired by the system. We explore how this curvature constrains the system's dynamical behavior, including its evolution speed and Fubini-Study distance between the quantum states. Within this geometric framework, we address the quantum brachistochrone problem and derive the minimal time required for optimal evolution, a result useful for time-efficient quantum circuit design. Subsequently, we explore the role of entanglement in shaping the state space geometry, modulating geometric phase, and controlling evolution speed and brachistochrone time. Our results reveal that entanglement enhances dynamics up to a critical threshold, beyond which geometric constraints begin to hinder evolution. Moreover, entanglement induces critical shifts in the geometric phase, making it a sensitive indicator of entanglement levels and a practical tool for steering quantum evolution. This approach offers valuable guidance for developing quantum technologies that require time-efficient control strategies rooted in the geometry of quantum state space.
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 manuscript explores the Riemannian geometry of quantum state manifolds and its physical implications. We establish that quantum state manifolds, as submanifolds of Hilbert space, possess intrinsic Riemannian geometry characterized by a metric tensor derived from the Hilbert space inner product. This geometric framework reveals fundamental connections: the distance between nearby states scales with quantum fluctuations, curvature indicates state robustness, and geometric phase transitions occur in many-body systems. We review historical developments from early quantum geometry to modern quantum information geometry, and present methodologies for computing geometric quantities and their physical interpretations. Results include distance-fluctuation relations for coherent states, curvature calculations for SU(2) manifolds, and applications to quantum optimization and metrology via the quantum Fisher information metric. The discussion interprets geometric quantities in terms of uncertainty principles, state robustness, and quantum criticality, while outlining future research directions in quantum chaos-geometry correlations, non-associative geometry, and geometric quantum algorithm design. We conclude that Riemannian geometry provides a powerful framework for understanding and exploiting the structure of quantum state space, with applications across quantum information processing, metrology, and foundational physics.
Geometric phase plays a fundamental role in quantum theory and accounts for wide phenomena ranging from the Aharanov-Bohm effect, the integer and fractional quantum hall effects, and topological phases of matter, including topological insulators, to name a few. In this thesis, we have proposed a fresh perspective of geodesics and null phase curves, which are key ingredients in understanding the geometric phase. We have also looked at a number of applications of geometric phases in topological phases, quantum walks, and non-inertial quantum systems. The shortest curve between any two points on a given surface is a (minimal) geodesic. They are also the curves along which a system does not acquire any geometric phase. In the same context, we can generalize geodesics to define a larger class of curves, known as null phase curves (NPCs), along which also the acquired geometric phase is zero; however, they need not be the shortest curves between the two points. We have proposed a geometrical decomposition of geodesics and null phase curves on the Bloch sphere, which is crucial in improving our understanding of the geometry of the state space and the intrinsic symmetries of geodesics and NPCs. We have also investigated the persistence of topological phases in quantum walks in the presence of an external (lossy) environment. We show that the topological order in one and two-dimensional quantum walks persist against moderate losses. Further, we use the geometric phase to detect the non-inertial modifications to the field correlators perceived by a circularly rotating two-level atom placed inside a cavity.
Quantum Geometry explores the intersection of probability theory and geometric analysis, aiming to establish a framework where quantum entanglement and geometric structure are intertwined. This paper investigates the potential of leveraging quantum geometry to provide a theoretical basis for quantum computing and quantum information. We propose a novel approach by framing quantum entanglement as geometric "lines" or "shapes," employing geometric operations to describe the evolution of quantum states. This investigation introduces a new perspective on probability theory and geometry, potentially offering a more intuitive and powerful way to understand quantum systems. The paper outlines the core concepts, mechanisms, and preliminary results of this research, highlighting its significance for advancing both quantum information theory and the fundamental understanding of quantum mechanics.