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Representation of Berry phase for spin system by the trajectories of Majorana stars

H. D. Liu, Libin Fu

Published Jun 26, 2014• 0 citations
OpenAlex

Abstract

The Majorana's stellar representation, which represents the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, provides an intuitive way to study a physical system with high dimensional projective Hilbert space. In this Letter, we study the Berry phase by these stars and their loops on the Bloch sphere. It is shown that the Berry phase of a general spin state can be expressed by an elegant formula with the solid angles of Majorana star loops. Furthermore, these results can be naturally used to a general state with arbitrary dimensions. To demonstrate our theory, we study a two mode interacting boson system. Finally, the relation between stars' correlations and quantum entanglement is discussed.

Related papers

Representation of Berry Phase by the Trajectories of Majorana Stars

H. D. Liu, Libin Fu

The Majorana's stellar representation, which represents the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, provides an intuitive way to study a physical system with a high dimensional projective Hilbert space. In this Letter, we study the Berry phase by these stars and their loops on the Bloch sphere. It is shown that the Berry phase of a general spin state can be expressed by an elegant formula with the solid angles of Majorana star loops. Furthermore, these results can be used to a general state with arbitrary dimensions. To demonstrate our theory, we study a two mode interacting boson system. Finally, the relation between stars' correlations and quantum entanglement is discussed.

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Berry phase and quantum entanglement in Majorana's stellar representation

H. D. Liu, Libin Fu

By presenting the evolution of a quantum state with the trajectories of the Majorana stars on the Bloch sphere, the Majorana's stellar provides an intuitive geometric picture to study a quantum system with high-dimensional Hilbert space. We study the Berry phase and quantum entanglement by distributions and motions of these stars on the Bloch sphere. It is shown that both of these unique characters of quantum state can be perfectly represented by the Majorana stars. The former is expressed by the solid angles of Majorana star loops and the distance between stars. For the latter, the distances between stars are also found to be a tool for measuring and classifying the multiparticle entanglement of a symmetric multiqubit pure state. To demonstrate our theory, we study a typical spin model which is equivalent to an interacting boson model or an interacting multiqubit system. The self-trapping phenomenon within is also discussed via the Majorana stars.

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Majorana stellar representation for mixed-spin (s, 1/2) systems*

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By describing the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, Majorana’s stellar representation provides an intuitive geometric perspective to comprehend the quantum system with high-dimensional Hilbert space. However, the representation of a two-spin coupling system on a Bloch sphere has not been solved satisfactorily yet. Here, a practical method is presented to resolve the problem for the mixed-spin ( s , 1/2) system and describe the entanglement of the system. The system can be decomposed into two spins: spin-( s + 1/2) and spin-( s – 1/2) at the coupling bases, which can be regarded as independent spins. Besides, any pure state may be written as a superposition of two orthonormal states with one spin-( s + 1/2) state and the other spin-( s – 1/2) state. Thus, the whole initial state can be regarded as a state of a pseudo spin-1/2. In this way, the mixed spin decomposes into three spins. Therefore, the state can be represented by (2 s + 1) + (2 s – 1) + 1 = 4 s + 1 sets of stars on a Bloch sphere. Finally, some examples are given to show symmetric patterns on the Bloch sphere and unveil the properties of the high-spin system by analyzing the trajectories of the Majorana stars on the Bloch sphere.

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Abstract Majorana stars, the antipodal directions associated with the coherent states that are orthogonal to a spin state, provide a visualization and a geometric understanding of the structures of general quantum states. For example, the Berry phase of a spin-1/2 is given by half the solid angle enclosed by the close path of its Majorana star. It is conceivable that the Berry phase of higher spins may also be related to the geometry of the Majorana constellation. We find that for a spin-1 state, besides the expected contributions from the solid angles enclosed by the close paths of the two Majorana stars, the Berry phase includes a term related to the twist of the relative position vector around the barycenter vector of the two Majorana stars, i.e., the self-rotation of the constellation. Interestingly, if the spin-1 state is taken as a symmetrized two-qubit state, the extra contribution to the Berry phase is given by the self-rotation of the Majorana constellation weighted by the quantum entanglement of the two qubits. This discovery alludes to the relevance of the Majorana stellar geometry in representing the deep structures of quantum states and of quantum entanglement.

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Majorana decomposition for two-qubit pure states

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Abstract The Majorana representation (MR), which represents a quantum state and its evolution with the Majorana stars (MSs) on the Bloch sphere, provides us an intuitive way to study symmetric multiqubit pure states physical system with SU (2) symmetry. In this work, we propose a method to extend the MR for generic two-qubit pure states. By adding an additional reference star (RS), which represents the symmetry of the qubits, on the symmetric two qubit state with two base stars, we give a more general MR for both symmetric, anti-symmetric and generic two-qubit states. Using this method, we study the entanglement, Berry phase and their relation by this Majorana decomposition. Furthermore, the symmetries for different type of two-qubit states are also discussed via the RSs and correlation between the two MSs.

Journal of Physics A Mathematical and Theoretical•Jun 3, 2021•2 citations
The Majorana's stellar representation, which represents the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, provides an intuitive way to study a physical system with high dimensional projective Hilbert space. In this Letter, we study the Berry phase by these stars and their loops on the Bloch sphere. It is shown that the Berry phase of a general spin state can be expressed by an elegant formula with the solid angles of Majorana star loops. Furthermore, these results can be naturally used to a general state with arbitrary dimensions. To demonstrate our theory, we study a two mode interacting boson system. Finally, the relation between stars' correlations and quantum entanglement is discussed.
The Majorana's stellar representation, which represents the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, provides an intuitive way to study a physical system with a high dimensional projective Hilbert space. In this Letter, we study the Berry phase by these stars and their loops on the Bloch sphere. It is shown that the Berry phase of a general spin state can be expressed by an elegant formula with the solid angles of Majorana star loops. Furthermore, these results can be used to a general state with arbitrary dimensions. To demonstrate our theory, we study a two mode interacting boson system. Finally, the relation between stars' correlations and quantum entanglement is discussed.
By presenting the evolution of a quantum state with the trajectories of the Majorana stars on the Bloch sphere, the Majorana's stellar provides an intuitive geometric picture to study a quantum system with high-dimensional Hilbert space. We study the Berry phase and quantum entanglement by distributions and motions of these stars on the Bloch sphere. It is shown that both of these unique characters of quantum state can be perfectly represented by the Majorana stars. The former is expressed by the solid angles of Majorana star loops and the distance between stars. For the latter, the distances between stars are also found to be a tool for measuring and classifying the multiparticle entanglement of a symmetric multiqubit pure state. To demonstrate our theory, we study a typical spin model which is equivalent to an interacting boson model or an interacting multiqubit system. The self-trapping phenomenon within is also discussed via the Majorana stars.
By describing the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, Majorana’s stellar representation provides an intuitive geometric perspective to comprehend the quantum system with high-dimensional Hilbert space. However, the representation of a two-spin coupling system on a Bloch sphere has not been solved satisfactorily yet. Here, a practical method is presented to resolve the problem for the mixed-spin ( s , 1/2) system and describe the entanglement of the system. The system can be decomposed into two spins: spin-( s + 1/2) and spin-( s – 1/2) at the coupling bases, which can be regarded as independent spins. Besides, any pure state may be written as a superposition of two orthonormal states with one spin-( s + 1/2) state and the other spin-( s – 1/2) state. Thus, the whole initial state can be regarded as a state of a pseudo spin-1/2. In this way, the mixed spin decomposes into three spins. Therefore, the state can be represented by (2 s + 1) + (2 s – 1) + 1 = 4 s + 1 sets of stars on a Bloch sphere. Finally, some examples are given to show symmetric patterns on the Bloch sphere and unveil the properties of the high-spin system by analyzing the trajectories of the Majorana stars on the Bloch sphere.
Abstract Majorana’s stellar representation provides an intuitive picture in which quantum states in high-dimensional Hilbert space can be observed using the trajectory of Majorana stars. We consider the Majorana’s stellar representation of the quantum geometric tensor for a spin state up to spin-3/2. The real and imaginary parts of the quantum geometric tensor, corresponding to the quantum metric tensor and Berry curvature, are therefore obtained in terms of the Majorana stars. Moreover, we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases. Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana’s stars.
Abstract Majorana stars, the antipodal directions associated with the coherent states that are orthogonal to a spin state, provide a visualization and a geometric understanding of the structures of general quantum states. For example, the Berry phase of a spin-1/2 is given by half the solid angle enclosed by the close path of its Majorana star. It is conceivable that the Berry phase of higher spins may also be related to the geometry of the Majorana constellation. We find that for a spin-1 state, besides the expected contributions from the solid angles enclosed by the close paths of the two Majorana stars, the Berry phase includes a term related to the twist of the relative position vector around the barycenter vector of the two Majorana stars, i.e., the self-rotation of the constellation. Interestingly, if the spin-1 state is taken as a symmetrized two-qubit state, the extra contribution to the Berry phase is given by the self-rotation of the Majorana constellation weighted by the quantum entanglement of the two qubits. This discovery alludes to the relevance of the Majorana stellar geometry in representing the deep structures of quantum states and of quantum entanglement.
Abstract The Majorana representation (MR), which represents a quantum state and its evolution with the Majorana stars (MSs) on the Bloch sphere, provides us an intuitive way to study symmetric multiqubit pure states physical system with SU (2) symmetry. In this work, we propose a method to extend the MR for generic two-qubit pure states. By adding an additional reference star (RS), which represents the symmetry of the qubits, on the symmetric two qubit state with two base stars, we give a more general MR for both symmetric, anti-symmetric and generic two-qubit states. Using this method, we study the entanglement, Berry phase and their relation by this Majorana decomposition. Furthermore, the symmetries for different type of two-qubit states are also discussed via the RSs and correlation between the two MSs.