When is there a multipartite maximum entangled state?

TL;DR

This paper investigates the existence of a maximum entangled state in multipartite quantum systems, establishing a necessary and sufficient condition based on the system dimensions, and employs algebraic complexity theory in the proof.

Contribution

It proves that a maximum entangled state exists if and only if the largest subsystem dimension is at least the product of the other dimensions, linking quantum information to algebraic complexity.

Findings

Maximum entangled state exists when $d_1 \\ge \\prod_{i=2}^n d_i$

The condition is both necessary and sufficient

Uses algebraic complexity theory in the proof

Abstract

For a multipartite quantum system of the dimension $d_1\otimes d_2\otimes... d_n$, $d_1\ge d_2\ge...\ge d_n$, is there an entangled state {\em maximum} in the sense that all other states in the system can be obtained from the state through local quantum operations and classical communications (LOCC)? When $d_1\ge\Pi_{i=2}^n d_i$, such state exists. We show that this condition is also necessary. Our proof, somewhat surprisingly, uses results from algebraic complexity theory.