Bounds on k-Uniform Quantum States

Do N-partite k-uniform states always exist when $k\leq \left \lfloor {{\frac {N}{2}}}\right \rfloor -1$ ? 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 code, 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 $\mathbb {C}^{d_{1}}\otimes (\mathbb {C}^{d_{2}})^{\otimes 2n}$ .