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arXiv:1601.07612 · Communications in Theoretical Physics

Coherent-State Approach for Majorana Representation*

Hao-Di 昊迪 Liu 刘, Libin Fu, Xiaoguang Wang

Published Jun 1, 2017• 6 citations
arXivHTMLPDF

Abstract

Abstract By representing a quantum state and its evolution with the Majorana stars on the Bloch sphere, the Majorana representation provides us an intuitive way to study a physical system with SU(2) symmetry. In this work, based on coherent states, we propose a method to establish the generalization of Majorana representation for a general symmetry. By choosing a generalized coherent state as a reference state, we give a more general Majorana representation for both finite and infinite systems and the corresponding star equations are given. Using this method, we study the squeezed vacuum states for three different symmetries, Heisenberg–Weyl, SU(2) and SU(1,1), and express the effect of squeezing parameter on the distribution of stars. Furthermore, we also study the dynamical evolution of stars for an initial coherent state driven by a nonlinear Hamiltonian, and find that at a special time point, the stars are distributed on two orthogonal large circles.

DOI
10.1088/0253-6102/67/6/611

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Abstract By representing a quantum state and its evolution with the Majorana stars on the Bloch sphere, the Majorana representation provides us an intuitive way to study a physical system with SU(2) symmetry. In this work, based on coherent states, we propose a method to establish the generalization of Majorana representation for a general symmetry. By choosing a generalized coherent state as a reference state, we give a more general Majorana representation for both finite and infinite systems and the corresponding star equations are given. Using this method, we study the squeezed vacuum states for three different symmetries, Heisenberg–Weyl, SU(2) and SU(1,1), and express the effect of squeezing parameter on the distribution of stars. Furthermore, we also study the dynamical evolution of stars for an initial coherent state driven by a nonlinear Hamiltonian, and find that at a special time point, the stars are distributed on two orthogonal large circles.
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.
Abstract The Majorana representation, which represents a quantum state by stars on the Bloch sphere, provides us an intuitive tool to study the quantum evolution in high dimensional Hilbert space. In this work, we investigate the second quantized model and the mean-field model for the interacting-boson system in the Majorana representation. It is shown that the motions of states in the two models are same in the linear case. Furthermore, the contribution of the nonlinear interaction to the star motions in the second quantized model can be expressed by a single star part which is equal to the nonlinear part of the equation for the star in mean-field model under large boson number limit and an extra part caused by the correlation between stars. These differences and relations can not only be reflected by the population differences between the two boson modes in the two models, but also lie with the differences between the continuous changes of the second quantized evolution with the nonlinear interacting strength and the critical behavior of the mean-field evolution which related to the self-trapping effect. The reason of the difference between the two models is also discussed by an effective Hamiltonian.
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.
Majorana stars, the 2S2S2S spin coherent states that are orthogonal to a spin-SSS state, offer an elegant method to visualize quantum states. This representation offers deep insights into the structure, symmetries, and entanglement properties of quantum states, bridging abstract algebraic formulations with intuitive geometrical intuition. In this paper, we briefly survey the development and applications of the Majorana constellation, exploring its relevance in modern areas of quantum information.
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.
While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2d>2d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.