Bounds on $k$-Uniform Quantum States

Abstract

Do $N$-partite $k$-uniform states always exist when $k\le \lfloor \frac{N}{2}\rfloor -1$? In this work, we provide new upper bounds on the parameter $k$ for the existence of $k$-uniform states in $({\mathbb{C}}^d{)}^{\otimes N}$ when $d=3,4,5$, which extend Rains' bound in 1999 and improve Scott's bound in 2004. Since a $k$-uniform state in $({\mathbb{C}}^d{)}^{\otimes N}$ corresponds to a pure $((N,1,k+1){)}_d$ quantum error-correcting codes, we also give new upper bounds on the minimum distance $k+1$ of pure $((N,1,k+1){)}_d$ quantum error-correcting codes. Furthermore, we generalize Scott's bound to heterogeneous systems, and show some non-existence results of absolutely maximally entangled states in ${\mathbb{C}}^{d_1}\otimes ({\mathbb{C}}^{d_2}{)}^{\otimes 2n}$.