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Physical Review A

Many-body anticommutator and its applications in the normalization of completely symmetric states with Majorana's stellar representations

Xi Lu, Liyao Zhan, Jie Chen, Hao-Di Liu, Libin Fu, Xiaoguang Wang

Published Jul 6, 2021• 2 citations
Publisher

Abstract

Understanding the quantum properties of many-body states is important in the fields of both quantum information and condensed-matter physics. For this purpose, we generalize the basic concept of the anticommutator from two operators to the many-body case. Some key properties are given for the many-body anticommutators, and examples are provided for Pauli operators and density matrices. Using these results and techniques from the symmetry group, in a straightforward way, we give the normalization of the completely symmetric states with Majorana's stellar representations. Our developed method will help to pave the way in the study of many-body symmetric systems.

DOI
10.1103/physreva.104.012203

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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)].
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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.
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