Extremal Maximal Entanglement

TL;DR

This paper investigates the maximum number of maximally mixed reductions in multipartite quantum states when absolute maximal entanglement is impossible, establishing bounds and exact values for specific cases using graph theory.

Contribution

It introduces a novel connection between quantum entanglement properties and Turán's problem in graph theory, providing bounds and exact values for $Qex(n)$.

Findings

Established a general upper bound for $Qex(n)$.

Provided lower bounds through constructive and probabilistic methods.

Determined that $Qex(8)=56$, a new exact value.

Abstract

A pure multipartite quantum state is called absolutely maximally entangled if all reductions of no more than half of the parties are maximally mixed. However, an $n$-qubit absolutely maximally entangled state only exists when $n$ equals $2$, $3$, $5$, and $6$. A natural question arises when it does not exist: which $n$-qubit pure state has the largest number of maximally mixed $\lfloor n/2 \rfloor$-party reductions? Denote this number by $Qex(n)$. It was shown that $Qex(4)=4$ in [Higuchi et al.Phys. Lett. A (2000)] and $Qex(7)=32$ in [Huber et al.Phys. Rev. Lett. (2017)]. In this paper, we give a general upper bound of $Qex(n)$ by linking the well-known Tur\'an's problem in graph theory, and provide lower bounds by constructive and probabilistic methods. In particular, we show that $Qex(8)=56$, which is the third known value for this problem.