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Properties of the Geometric Phases on Qbits

Anthony Pinedo Araujo

Published Jul 1, 2009• 0 citations
OpenAlex

Abstract

Since Berry demostrated that the standard description of adiabatic processes in quantum mechanics was incomplete, geometric phases have been studied in many areas of physics. Both adiabatic and non-adiabatic phase are described in detail, with the mathematical background. Then we study the qbit, that is the principal unit of information of the quantum computation, and its representation on the Bloch sphere. Finally we find the general expression for geometric phases for qbits. Those final expression are in fact related to the solid angle enclosed in the circuit on the Bloch sphere.

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Since Berry demostrated that the standard description of adiabatic processes in quantum mechanics was incomplete, geometric phases have been studied in many areas of physics. Both adiabatic and non-adiabatic phase are described in detail, with the mathematical background. Then we study the qbit, that is the principal unit of information of the quantum computation, and its representation on the Bloch sphere. Finally we find the general expression for geometric phases for qbits. Those final expression are in fact related to the solid angle enclosed in the circuit on the Bloch sphere.
The geometric phase in quantum mechanics is formulated for charged particles in a gauge-invariant, geometric manner. It is then extended to an evolution resulting from a sequence of measurements as in the work of Pancharatnam and Aharonov and Vardi. Its close connection to the Feynman formulation of quantum mechanics is pointed out. The geometric angles, which are generalizations of the classical, adiabatic angles introduced by Hannay and the quantum, adiabatic angles introduced by Anandan and Stodolsky in their group-theoretic treatment of Berry's phase, are studied in quantum and classical physics. The geometric phase for a quantum spin in a magnetic field due to a second particle is obtained using the quantum reference frame defined by the latter. The question of whether the geometric phase and angles are local or nonlocal and their relationship to the electromagnetic and gravitational phases are also discussed.
Non-Abelian and non-adiabatic variants of Berry's geometric phase have been pivotal in the recent advances in fault tolerant quantum computation gates, while Berry's phase itself is at the heart of the study of topological phases of matter. Here we use ultracold atoms to study the unique properties of spin-1 geometric phase. The spin vector of a spin-1 system, unlike that of a spin-1/2 system, can lie anywhere on or inside the Bloch sphere representing the phase space. This suggests a generalization of Berry's phase to include closed paths that go inside the Bloch sphere. Under this generalization, the special class of loops that pass through the center, which we refer to as\\textit {singular loops}, are significant in two ways. First, their geometric phase is non-Abelian and second, their geometrical properties are qualitatively different from the nearby non-singular loops, making them akin to critical points of a quantum phase transition. Here we use coherent control of ultracold Rb atoms in an optical trap to experimentally explore the geometric phase of singular loops in a spin-1 quantum system.
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 computing leverages fundamental principles of quantum mechanics to perform computations far exceeding classical architectures. At the core of single-qubit quantum state manipulation is the Bloch sphere, a geometric representation mapping the state space of two-level quantum systems onto a three-dimensional unit sphere. This paper presents a comprehensive theoretical framework of qubits, starting from single-qubit geometric state vectors and expanding to rigorous multi-qubit mathematical formulations. Furthermore, it analyzes single-qubit quantum gates as unitary rotations on the Bloch sphere, evaluates multi-qubit system scaling, and explicitly proves state normalization conditions using Born's rule across multi-dimensional Hilbert spaces.
The properties of the geometric phases between three quantum states are investigated in a high-dimensional Hilbert space using the Majorana representation of symmetric quantum states. We found that the geometric phases between the three quantum states in an NNN-state quantum system can be represented by $N\ensuremath{-}1$ spherical triangles on the Bloch sphere. The parameter dependence of the geometric phase was analyzed based on this picture. We found that the geometric phase exhibits rich nonlinear behavior in a high-dimensional Hilbert space.