Anticoherence and entanglement of spin states

Jeffrey Scott Pratt

Jason Crann, Rajesh Pereira, David W. Kribs

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.

Meichao Wang, Yan Mei Zhu

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.

K. Sengupta

Rosario Fazio, Chiara Macchiavello

Rajesh Pereira, Connor Paddock

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.