Extremal maximal entanglement

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 exists only 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 $\ensuremath{\lfloor}n/2\ensuremath{\rfloor}$-party reductions? We denote this number by $\mathrm{Qex}(n)$. It was shown that $\mathrm{Qex}(4)=4$ by Higuchi and Sudbery [Phys. Lett. A 273, 213 (2000)] and $\mathrm{Qex}(7)=32$ by Huber et al. [Phys. Rev. Lett. 118, 200502 (2017)]. In this paper, we give a general upper bound of $\mathrm{Qex}(n)$ by linking the well-known Tur\'an problem in graph theory and provide lower bounds using constructive and probabilistic methods. In particular, we show that $\mathrm{Qex}(8)=56$, which is the third known value for this problem.