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:1707.02601

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

P. K. Aravind

Published Jul 9, 2017• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1707.02601

Related papers

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

Majorana Treatment of Spin-1 System

Tenzin Kalden

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.

•Jan 1, 2015•0 citations

SIC-POVMS and MUBS: Geometrical Relationships in Prime Dimension

D. M. Appleby, Luigi Accardi, Guillaume Adenier +5 more

The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d^2-1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d+1 mutually orthogonal d-1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d+1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs.

arXiv:0905.1428•AIP conference proceedings•Jan 1, 2009•24 citations

SIC-POVMS AND MUBS: GEOMETRICAL RELATIONSHIPS IN PRIME DIMENSION

D. M. Appleby

The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d 2 − 1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d + 1 mutually orthogonal d − 1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d + 1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs. 1

•Dec 4, 2012•40 citations

Majorana stellar representation for mixed-spin (s, 1/2) systems*

Yuguo Su, Fei Yao, Hongbin Liang +3 more

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.

arXiv:1904.02462•Chinese Physics B•Oct 20, 2020•3 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
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.
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.
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d^2-1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d+1 mutually orthogonal d-1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d+1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs.
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d 2 − 1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d + 1 mutually orthogonal d − 1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d + 1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs. 1
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.