Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1902.00684 · Physical Review A

Three-tangle of a general three-qubit state in the representation of Majorana stars

Chon‐Fai Kam, Ren‐Bao Liu

Published Mar 13, 2020• 9 citations
arXivHTMLPDF

Abstract

Majorana stars, the $2j$ spin coherent states that are orthogonal to a spin-$j$ state, offer a visualization of general quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine tripartite entanglement---the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state---is measured by the normalized product of the distance between the Majorana stars. However, the Majorana representation cannot be applied to general nonsymmetric $n$-qubit states. We show that, after a series of $\mathrm{SL}(2,\mathbb{C})$ transformations, nonsymmetric three-qubit states can be transformed to symmetric three-qubit states, while at the same time the three-tangle is unchanged. Thus the genuine tripartite entanglement of general three-qubit states has the geometric representation of the associated Majorana stars. The symmetrization and hence the Majorana star representation of certain genuine high-order entanglement for more qubits are possible for some special states. In general cases, however, the constraints on the symmetrization may prevent the Majorana star representation of the genuine entanglement.

DOI
10.1103/physreva.101.032318

Related papers

Berry phases of higher spins due to internal geometry of Majorana constellation and relation to quantum entanglement

Chon‐Fai Kam, Ren-Bao Liu

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.

arXiv:2008.06965•New Journal of Physics•Jun 25, 2021•4 citations

The quantum sky of Majorana stars

L. L. Sanchez-Soto, A. B. Klimov, A. Z. Goldberg +1 more

Majorana stars, the $2S$ spin coherent states that are orthogonal to a spin-$S$ 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.

quant-pharXiv:2601.20922•Jan 30, 2026•0 citations

Visualizing Three-Qubit Entanglement

Alfred Benedito, Germán Sierra

We present a graphical framework to represent entanglement in three-qubit states. The geometry associated with each entanglement class and type is analyzed, revealing distinct structural features. We explore the connection between this geometric perspective and the tangle, deriving bounds that depend on the entanglement class. Based on these insights, we conjecture a purely geometric expression for both the tangle and Cayley’s hyperdeterminant for non-generic states. As an application, we analyze the energy eigenstates of physical Hamiltonians, identifying the sufficient conditions for genuine tripartite entanglement to be robust under symmetry-breaking perturbations and level repulsion effects.

arXiv:2505.23638•Entropy•Jul 27, 2025•0 citations

Majorana Constellations: A Geometric Lens on Multipartite Entanglement and Geometric Phases

Chon-Fai Kam

The Majorana stellar representation maps a pure spin-$S$ state to $2S$ points on a sphere. This review develops it with entanglement as the organising principle, and two objects recur throughout: the constellation, and the permanent of the Gram matrix of its stars. The degeneracy pattern of the constellation is invariant under stochastic local operations and classical communication, so the integer partitions of $N$ label a finite set of families of symmetric $N$-qubit states. That labelling is a coarse-graining rather than a classification, since from four distinct stars onwards each family carries continuous Möbius moduli. The permanent supplies what the pattern omits. Normalised by it, inter-star chordal distances give the concurrence and the three-tangle in closed form, and the same permanent governs the anomalous contribution to the Berry phase acquired under adiabatic cyclic evolution, so that a single quantity links static correlations to dynamical holonomy. We also fix the computational reach of the geometry. Overlaps of symmetric states are permanents of matrices of rank at most two and are polynomially computable, whereas measures defined by an optimisation over the sphere are not reached by that argument. Interest in stellar representations has resurged, but the literature remains dispersed, and no existing treatment develops the link between constellation geometry, multipartite entanglement, and geometric phases within a single framework. The same two objects organise the applications reviewed here, from extremal states in metrology and permutation-invariant codes to collective spin models and photonic constellations, together with extensions to mixed states and to continuous-variable systems through the stellar rank. Whether the anomalous phase admits a bound in terms of any entanglement monotone remains open.

quant-pharXiv:2605.15008•Physica Scripta•May 15, 2026•0 citations

Coherent-State Approach for Majorana Representation*

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

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.

arXiv:1601.07612•Communications in Theoretical Physics•Jun 1, 2017•6 citations

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.

Physical Review A•Aug 31, 2016•34 citations
Majorana stars, the 2j2j2j spin coherent states that are orthogonal to a spin-jjj state, offer a visualization of general quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine tripartite entanglement---the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state---is measured by the normalized product of the distance between the Majorana stars. However, the Majorana representation cannot be applied to general nonsymmetric nnn-qubit states. We show that, after a series of SL(2,C)\mathrm{SL}(2,\mathbb{C})SL(2,C) transformations, nonsymmetric three-qubit states can be transformed to symmetric three-qubit states, while at the same time the three-tangle is unchanged. Thus the genuine tripartite entanglement of general three-qubit states has the geometric representation of the associated Majorana stars. The symmetrization and hence the Majorana star representation of certain genuine high-order entanglement for more qubits are possible for some special states. In general cases, however, the constraints on the symmetrization may prevent the Majorana star representation of the genuine entanglement.
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.
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.
We present a graphical framework to represent entanglement in three-qubit states. The geometry associated with each entanglement class and type is analyzed, revealing distinct structural features. We explore the connection between this geometric perspective and the tangle, deriving bounds that depend on the entanglement class. Based on these insights, we conjecture a purely geometric expression for both the tangle and Cayley’s hyperdeterminant for non-generic states. As an application, we analyze the energy eigenstates of physical Hamiltonians, identifying the sufficient conditions for genuine tripartite entanglement to be robust under symmetry-breaking perturbations and level repulsion effects.
The Majorana stellar representation maps a pure spin-SSS state to 2S2S2S points on a sphere. This review develops it with entanglement as the organising principle, and two objects recur throughout: the constellation, and the permanent of the Gram matrix of its stars. The degeneracy pattern of the constellation is invariant under stochastic local operations and classical communication, so the integer partitions of NNN label a finite set of families of symmetric NNN-qubit states. That labelling is a coarse-graining rather than a classification, since from four distinct stars onwards each family carries continuous Möbius moduli. The permanent supplies what the pattern omits. Normalised by it, inter-star chordal distances give the concurrence and the three-tangle in closed form, and the same permanent governs the anomalous contribution to the Berry phase acquired under adiabatic cyclic evolution, so that a single quantity links static correlations to dynamical holonomy. We also fix the computational reach of the geometry. Overlaps of symmetric states are permanents of matrices of rank at most two and are polynomially computable, whereas measures defined by an optimisation over the sphere are not reached by that argument. Interest in stellar representations has resurged, but the literature remains dispersed, and no existing treatment develops the link between constellation geometry, multipartite entanglement, and geometric phases within a single framework. The same two objects organise the applications reviewed here, from extremal states in metrology and permutation-invariant codes to collective spin models and photonic constellations, together with extensions to mixed states and to continuous-variable systems through the stellar rank. Whether the anomalous phase admits a bound in terms of any entanglement monotone remains open.
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.