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 introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.

Abstract. Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group G and -injectivity of L ∞ (Ĝ) as an operator L (Ĝ)-module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra V N(G) is -injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded L (Ĝ)-module maps on L ∞ (Ĝ), and give a simpli ed proof that amenable discrete quantum groups have co-amenable compact duals which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.