Entanglement and Designs

TL;DR

This paper explores the relationship between entanglement and conical 2-designs, generalizing known measurement frameworks and linking them to entanglement monotones like concurrence.

Contribution

It establishes a characterization of conical 2-designs via entanglement monotones and extends previous work on designs and entanglement detection.

Findings

Conical 2-designs are equivalent to the existence of certain entanglement monotones.

The concurrence is a regular entanglement monotone related to conical 2-designs.

Generalizations of designs facilitate entanglement detection methods.

Abstract

We describe a connection between entanglement and designs. It involves the conical 2-designs introduced in a previous paper. These are a generalization of projective 2-designs which includes full sets of arbitrary rank mutually unbiased measurements (MUMs) and arbitrary rank symmetric informationally complete measurements (SIMs), as well as the more familiar MUBs and SICs. We show that a POVM is a conical 2-design if and only if there exists what we call a regular entanglement monotone whose restriction to the pure states is a function of the norm of the probability vector. In that case the concurrence is such a monotone. We also generalize and develop previous work on designs and entanglement detection.