Majorana Star and Invariant Vector Representations of a Single Qutrit

The Qutrit state density matrix is of order 3 and depends on 8 parameters in general case. Visualization of this 8-dimensional state space is practically impossible using 8-dimensional vectors commonly used. Recently a 3-dimensional vector representation of the Qutrit state space (also called Invariant Vector Representation) has been proposed [1]. In this work we present the time-evolution of the IVR vectors for the Qutrit Cascade or Ξ-model to emphasize the advantages of the IVR and also relate IVR vectors to well-known stars of the Majorana State Representation (MSR).The Qutrit state density matrix is of order 3 and depends on 8 parameters in general case. Visualization of this 8-dimensional state space is practically impossible using 8-dimensional vectors commonly used. Recently a 3-dimensional vector representation of the Qutrit state space (also called Invariant Vector Representation) has been proposed [1]. In this work we present the time-evolution of the IVR vectors for the Qutrit Cascade or Ξ-model to emphasize the advantages of the IVR and also relate IVR vectors to well-known stars of the Majorana State Representation (MSR).

arXiv:1910.03664Oct 8, 20190 citations

Majorana representation of symmetric multiqubit states

A. R. Usha Devi, Sudha, A. K. Rajagopal

arXiv:1103.3640Quantum Information ProcessingAug 26, 201170 citations

Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates

Shruti Dogra, Kavita Dorai, Arvind

Abstract We report a study of the Majorana geometrical representation of a qutrit, where a pair of points on a unit sphere represents its quantum states. A canonical form for qutrit states is presented, where every state can be obtained from a one-parameter family of states via SO (3) action. The notion of spin-1 magnetization which is invariant under SO (3) is geometrically interpreted on the Majorana sphere. Furthermore, we describe the action of several quantum gates in the Majorana picture and experimentally implement these gates on a spin-1 system (an NMR qutrit) oriented in a liquid crystalline environment. We study the dynamics of the pair of points representing a qutrit state under various useful quantum operations and connect them to different NMR operations. Finally, using the Gell Mann matrix picture we experimentally implement a scheme for complete qutrit state tomography.Abstract We report a study of the Majorana geometrical representation of a qutrit, where a pair of points on a unit sphere represents its quantum states. A canonical form for qutrit states is presented, where every state can be obtained from a one-parameter family of states via SO (3) action. The notion of spin-1 magnetization which is invariant under SO (3) is geometrically interpreted on the Majorana sphere. Furthermore, we describe the action of several quantum gates in the Majorana picture and experimentally implement these gates on a spin-1 system (an NMR qutrit) oriented in a liquid crystalline environment. We study the dynamics of the pair of points representing a qutrit state under various useful quantum operations and connect them to different NMR operations. Finally, using the Gell Mann matrix picture we experimentally implement a scheme for complete qutrit state tomography.

arXiv:1703.06102Journal of Physics B Atomic Molecular and Optical PhysicsJan 10, 201815 citations

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

Chon‐Fai Kam, Ren‐Bao Liu

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.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.

arXiv:1902.00684Physical Review AMar 13, 20209 citations

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.