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Majorana Treatment of Spin-1 System

Tenzin Kalden

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

The objective of this project is to use the Majorana description of a spin-1 system to examine many of its properties. In the Majorana approach, an arbitrary pure state of a spin-1 system is represented by a pair of points on the unit sphere, or a pair of unit vectors. An expression is derived for the squared overlap of two states in terms of their Majorana (or M-) vectors and used to work out all the orthogonalities and bases that are possible for the three geometrically distinct types of spin-1 states. Finally, a geometrical construction is given of a set of mutually unbiased bases for a spin-1 system. The applications of this result to problems in quantum information are mentioned.

Related papers

MUBs and SIC-POVMs of a spin-1 system from the Majorana approach

P. K. Aravind

In the Majorana or stellar representation of quantum states, an arbitrary (pure) state of a spin-1 system is represented by a pair of points on the unit sphere or, equivalently, by a pair of unit vectors. This paper presents an expression for the squared modulus of the inner product of two spin-1 states in terms of their Majorana vectors and uses it to give a geometrical construction of the MUBs and SIC-POVMs of a spin-1 system. The results are not new and duplicate those obtained earlier by other methods, but the Majorana approach nevertheless illuminates them from an unusual point of view. In particular, it reveals the MUBs and SICs as symmetrical collections of vectors in ordinary three-dimensional space, rather than as rays in a projective Hilbert space. While it does not appear feasible to extend this treatment to higher spin systems, the spin-1 case exhibits sufficient subtlety and complexity to be worth spelling out for its pedagogical and historical interest.

arXiv:1707.02601•Jul 9, 2017•3 citations

Geometrical Construction of MUBS and SIC-POVMS for Spin-1 Systems

Tenzin Kalden

The objective of this thesis is to use the Majorana description of a spin-1 system to give a geometrical construction of a maximal set of Mutually Unbiased Bases (MUBs) and Symmetric Informationally Complete Positive Operator Valued Measures (SIC-POVMs) for this system. In the Majorana Approach, an arbitrary pure state of a spin-1 system is represented by a pair of points on the Reimann sphere, or a pair of unit vectors (known as Majorana vectors or M-vectors). Spin-1 states can be of three types: those whose vectors are parallel, those whose vectors are antiparallel and those whose vectors make an arbitrary angle. The types of bases possible for a spin-1 system are thus geometrically much more varied than for a spin-half system or qubit, which is the standard unit of information storage in most quantum protocols. Our derivation of the MUBs and SIC-POVMs proceeds from a recently derived expression for the squared overlap of two spin-1 states in terms of their M-vectors and the minimal additional set of assumptions that are needed. These assumptions include time-reversal invariance in the case of the MUBs and the requirement of three-fold symmetry in the case of the SIC-POVMs. The applications of these results to problems in quantum information are mentioned.

•Apr 28, 2016•0 citations

From Majorana to Bloch: two ways to generalize the Bloch sphere representation

Louis Hanotel

While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states ($d>2$) 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.

quant-pharXiv:2610.11868•Oct 9, 2026•0 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.

arXiv:1703.06102•Journal of Physics B Atomic Molecular and Optical Physics•Jan 10, 2018•15 citations

Majorana representation for mixed states

Eduardo Serrano-Ensástiga, Daniel Braun

We generalize the Majorana stellar representation of spin-$s$ pure states to mixed states, and in general to any Hermitian operator, defining a bijective correspondence between three spaces: the spin-density matrices, a projective space of homogeneous polynomials of four variables, and a set of equivalence classes of points (constellations) on spheres of different radii. The representation behaves well under rotations by construction, and also under partial traces where the reduced density matrices inherit their constellation classes from the original state $\ensuremath{\rho}$. We express several concepts and operations related to density matrices in terms of the corresponding polynomials, such as the anticoherence criterion and the tensor representation of spin-$s$ states described in Giraud et al. [Phys. Rev. Lett. 114, 080401 (2015)].

arXiv:1909.07740•Physical Review A•Feb 26, 2020•25 citations

A Quantum Constellation

Qian Niu

Majorana's geometrical representation of quantum spin as points on a sphere offers an intuitive approach to understanding quantum systems with multiple components.

Physics•Jun 11, 2012•9 citations
The objective of this project is to use the Majorana description of a spin-1 system to examine many of its properties. In the Majorana approach, an arbitrary pure state of a spin-1 system is represented by a pair of points on the unit sphere, or a pair of unit vectors. An expression is derived for the squared overlap of two states in terms of their Majorana (or M-) vectors and used to work out all the orthogonalities and bases that are possible for the three geometrically distinct types of spin-1 states. Finally, a geometrical construction is given of a set of mutually unbiased bases for a spin-1 system. The applications of this result to problems in quantum information are mentioned.
In the Majorana or stellar representation of quantum states, an arbitrary (pure) state of a spin-1 system is represented by a pair of points on the unit sphere or, equivalently, by a pair of unit vectors. This paper presents an expression for the squared modulus of the inner product of two spin-1 states in terms of their Majorana vectors and uses it to give a geometrical construction of the MUBs and SIC-POVMs of a spin-1 system. The results are not new and duplicate those obtained earlier by other methods, but the Majorana approach nevertheless illuminates them from an unusual point of view. In particular, it reveals the MUBs and SICs as symmetrical collections of vectors in ordinary three-dimensional space, rather than as rays in a projective Hilbert space. While it does not appear feasible to extend this treatment to higher spin systems, the spin-1 case exhibits sufficient subtlety and complexity to be worth spelling out for its pedagogical and historical interest.
The objective of this thesis is to use the Majorana description of a spin-1 system to give a geometrical construction of a maximal set of Mutually Unbiased Bases (MUBs) and Symmetric Informationally Complete Positive Operator Valued Measures (SIC-POVMs) for this system. In the Majorana Approach, an arbitrary pure state of a spin-1 system is represented by a pair of points on the Reimann sphere, or a pair of unit vectors (known as Majorana vectors or M-vectors). Spin-1 states can be of three types: those whose vectors are parallel, those whose vectors are antiparallel and those whose vectors make an arbitrary angle. The types of bases possible for a spin-1 system are thus geometrically much more varied than for a spin-half system or qubit, which is the standard unit of information storage in most quantum protocols. Our derivation of the MUBs and SIC-POVMs proceeds from a recently derived expression for the squared overlap of two spin-1 states in terms of their M-vectors and the minimal additional set of assumptions that are needed. These assumptions include time-reversal invariance in the case of the MUBs and the requirement of three-fold symmetry in the case of the SIC-POVMs. The applications of these results to problems in quantum information are mentioned.
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.
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.
We generalize the Majorana stellar representation of spin-sss pure states to mixed states, and in general to any Hermitian operator, defining a bijective correspondence between three spaces: the spin-density matrices, a projective space of homogeneous polynomials of four variables, and a set of equivalence classes of points (constellations) on spheres of different radii. The representation behaves well under rotations by construction, and also under partial traces where the reduced density matrices inherit their constellation classes from the original state $\ensuremath{\rho}$. We express several concepts and operations related to density matrices in terms of the corresponding polynomials, such as the anticoherence criterion and the tensor representation of spin-sss states described in Giraud et al. [Phys. Rev. Lett. 114, 080401 (2015)].