A note on the Bloch representation of absolutely maximally entangled states

TL;DR

This paper derives an explicit Bloch representation for AME states across various dimensions and subsystem counts, revealing properties of their support trace and establishing conditions for their existence.

Contribution

It provides a generalized analytic formula for the Bloch representation of AME states, linking support trace to hypergeometric functions and identifying optimal conditions for their existence.

Findings

Support trace is given by hypergeometric functions, independent of subsystem choices.

Derived an explicit formula for the Bloch representation of AME states.

Identified the optimal conditions for the existence of AME states.

Abstract

The absolutely maximally entangled (AME) states play key roles in quantum information processing. We provide an explicit expression of the generalized Bloch representation of AME states for general dimension $d$ of individual subsystems and arbitrary number of partite $n$. Based on this analytic formula, we prove that the trace of the squared support for any given weight is given by the so-called hyper-geometric function and is irrelevant with the choices of the subsystems. The optimal point for the existence of AME states is obtained.