No abstract

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 examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.

We introduce and study the l 1 norm of coherence of assistance both theoretically and operationally. We first provide an upper bound for the l 1 norm of coherence of assistance and show a necessary and sufficient condition for the saturation of the upper bound. For two and three dimensional quantum states, the analytical expression of the l 1 norm of coherence of assistance is given.Operationally, the mixed quantum coherence can always be increased with the help of another party's local measurement and one way classical communication since the l 1 norm of coherence of assistance, as well as the relative entropy of coherence of assistance, is shown to be strictly larger than the original coherence. The relation between the l 1 norm of coherence of assistance and entanglement is revealed. Finally, a comparison between the l 1 norm of coherence of assistance and the relative entropy of coherence of assistance is made.

Abstract. We conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one-way local operations and classical communications (LOCC). We show how operator structures such as operator systems, operator algebras, and Hilbert C * -modules all naturally arise in this setting, and we make use of these structures to derive new results and new derivations of some established results in the study of LOCC. We also show that perfect distinguishability under oneway LOCC and under arbitrary operations is equivalent for several families of operators that appear jointly in matrix and operator theory and quantum information theory.