Quantum Conical Designs

TL;DR

This paper introduces conical t-designs, a generalization of complex projective t-designs, encompassing arbitrary rank SICs and MUMs, with implications for quantum entanglement and polytope structures in the Bloch body.

Contribution

It defines conical t-designs, characterizes homogeneous conical 2-designs, and links them to symmetric polytopes, opening avenues for discovering new projective 2-designs.

Findings

Conical t-designs include arbitrary rank SICs and MUMs.

Homogeneous conical 2-designs are characterized by specific polytope conditions.

Conical 2-designs relate to symmetric decompositions of Werner states.

Abstract

Complex projective t-designs, particularly SICs and full sets of MUBs, play an important role in quantum information. We introduce a generalization which we call conical t-designs. They include arbitrary rank symmetric informationally complete measurements (SIMs) and full sets of arbitrary rank mutually unbiased measurements (MUMs). They are deeply implicated in the description of entanglement (as we show in a subsequent paper). Viewed in one way a conical 2-design is a symmetric decomposition of a separable Werner state (up to a normalization factor). Viewed in another way it is a certain kind of polytope in the Bloch body. In the Bloch body picture SIMs and full sets of MUMs form highly symmetric polytopes (a single regular simplex in the one case; the convex hull of a set of orthogonal regular simplices in the other). We give the necessary and sufficient conditions for an arbitrary polytope to be what we call a homogeneous conical 2-design. This suggests a way to search for new kinds of projective 2-design.