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Journal of Physics A Mathematical and Theoretical

Majorana decomposition for two-qubit pure states

C Yang, H. D. Liu, Libin Fu

Published Jun 3, 2021• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/ac07eb

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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.
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.
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.
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.
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 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.
The completely symmetric states play an essential role in quantum physics. In this paper, we calculate the reduced density matrix (RDM) for a single particle of the completely symmetric system coupled by NNN spin-12\frac{1}{2}21​ particles, because it helps to investigate the evolution of expectation value for the observable and to calculate the entanglement between the subsystems. Furthermore, we use Majorana's stellar representation (MSR) to represent the results because it provides an intuitive geometric perspective to comprehend the quantum states in the high-dimensional Hilbert space with distributions and trajectories of the Majorana stars on a Bloch sphere. With the operation properties of the generalized many-body anticommutator, we get the general MSR form of a single-qubit RDM. As the application and verification, we calculate the single-qubit RDM for the Dicke states with the results. Similarly, we further solve the RDM of the spin-N2\frac{N}{2}2N​ state in a uniform magnetic field and study the systems with symmetric structures on the Bloch sphere. The results exhibit the relations between the composite systems and the subsystems, and provide a new idea for the numerical solution of multiqubit systems.