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Journal of Physics A Mathematical and Theoretical

Anticoherent spin-2 states and spherical designs

Meichao Wang, Yan Mei Zhu

Published Oct 3, 2022• 1 citation
Publisher

Abstract

Abstract Anticoherent spin states are introduced as quantum states that exhibit maximally nonclassical behaviors which are important in the study of quantum entanglement. Using Majorana representation of spin states, Crann–Pereira–Kribs studied the relation between anticoherent spin states and spherical designs. They proposed a conjecture that a spin- s state is anticoherent to order t if and only if its Majorana representation is a spherical t -design on S 2 . In this paper, we prove that this conjecture is true for s = 2.

DOI
10.1088/1751-8121/ac971d

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Abstract Anticoherent spin states are introduced as quantum states that exhibit maximally nonclassical behaviors which are important in the study of quantum entanglement. Using Majorana representation of spin states, Crann–Pereira–Kribs studied the relation between anticoherent spin states and spherical designs. They proposed a conjecture that a spin- s state is anticoherent to order t if and only if its Majorana representation is a spherical t -design on S 2 . In this paper, we prove that this conjecture is true for s = 2.
Anticoherent spin states are quantum states that exhibit maximally nonclassical behaviour in a certain sense. Any spin state whose Majorana representation is a Platonic solid is called a perfect state. By direct calculation, it has been shown that any perfect state is an anticoherent spin state. We show that any spin state whose Majorana representation is both the orbit of a finite subgroup of O (3) and a spherical t -design must be an anticoherent spin state of order t . Since all Platonic solids are spherical designs, this result gives an explanation of the anticoherence of perfect states and explains their observed order. We also show that any spin state whose Majorana representation lies in any single open hemisphere cannot be anticoherent of any order. This result is then used to give further relations between spherical designs and anticoherent spin states. We also pose some questions relating spherical designs and geometric entanglement.
We extend the notion of anticoherent spin states to anticoherent subspaces. An anticoherent subspace of order t is a subspace whose unit vectors are all anticoherent states of order at least t. We use Klein’s description of algebras of polynomials which are invariant under finite subgroups of SU(2) on C2 to provide constructions of anticoherent subspaces. We discuss applications of this idea to the entanglement of n qubit symmetric states. Furthermore, we show a connection between the existence of these subspaces and the properties of the higher-rank numerical range for certain products of spin observables. We also note that these constructions give us subspaces of spin states all of whose unit vectors have Majorana representations which are spherical designs of order at least t.
The set of pure spin states with vanishing spin expectation value can be regarded as the set of the less coherent pure spin states. This set can be divided into a finite number of nested subsets on the basis of higher order moments of the spin operators. This subdivision relies on the notion of anticoherent spin state to order ttt: A spin state is said to be anticoherent to order ttt if the moment of order kkk of the spin components along any directions are equal for k=1,2,...,tk=1,2,...,tk=1,2,...,t. Most spin states are neither coherent nor anticoherent, but can be arbitrary close to one or the other. In order to quantify the degree of anticoherence of pure spin states, we introduce the notion of anticoherence measures. By relying on the mapping between spin-jjj states and symmetric states of 2j2j2j spin 1/21/21/2 (Majorana representation), we present a systematic way of constructing anticoherence measures to any order. We briefly discuss their connection with measures of quantum coherence. Finally, we illustrate our measures on various spin states and use them to investigate the problem of the existence of anticoherent spin states with degenerated Majorana points.
We investigate multiqubit permutation-symmetric states with maximal entropy of entanglement. Such states can be viewed as particular spin states, namely anticoherent spin states. Using the Majorana representation of spin states in terms of points on the unit sphere, we analyze the consequences of a point-group symmetry in their arrangement on the quantum properties of the corresponding state. We focus on the identification of anticoherent states (for which all reduced density matrices in the symmetric subspace are maximally mixed) associated with point-group-symmetric sets of points. We provide three different characterizations of anticoherence and establish a link between point symmetries, anticoherence, and classes of states equivalent through stochastic local operations with classical communication. We then investigate in detail the case of small numbers of qubits and construct infinite families of anticoherent states with point-group symmetry of their Majorana points, showing that anticoherent states do exist to arbitrary order.
Anticoherent spin states have isotropic low-order spin moments and are relevant to direction-independent metrology and quantum reference-frame alignment. In contrast to pure states, for mixed states such isotropy may originate either from genuine quantum correlations or from classical statistical mixing. We introduce an axiomatic framework for mixed-state ttt-anticoherence based on the symmetric qubit embedding. We distinguish total ttt-anticoherence, non-decreasing under SU(2)-covariant channels, from quantum ttt-anticoherence, defined as a resource monotone relative to a chosen total measure and constrained to coincide with it on pure states. This yields a classical contribution as their difference. We construct total measures based on reduced-state purity, Hilbert-Schmidt distance, and cumulative multipoles, and we discuss fidelity-based total candidates. We construct quantum counterparts via convex-roof extensions of pure-state functionals tied to bipartite entanglement in the symmetric sector. We provide explicit mixed-state examples, identify states with maximal quantum anticoherence supported on anticoherent subspaces, study robustness under particle loss for different types of states, and characterize the trade-off between purity and the maximal achievable anticoherence order.