Bounds on k-Uniform Quantum States

Abstract

<p>Do N -partite k -uniform states always exist when k ≤ ⎣ N /2⎦ − 1? In this work, we provide new upper bounds on the parameter k for the existence of k -uniform states in (C ^<em>d</em> ) ^{⊗<em>N</em>} when d = 3, 4, 5, which extend Rains bound in 1999 and improve Scotts bound in 2004. Since a k -uniform state in (C ^<em>d</em> ) ^{⊗<em>N</em>} corresponds to a pure (( N , 1, k + 1)) _<em>d</em> quantum error-correcting code, we also give new upper bounds on the minimum distance k +1 of pure (( N , 1, k +1)) _<em>d</em> quantum errorcorrecting codes. Furthermore, we generalize Scotts bound to heterogeneous systems, and show some non-existence results of absolutely maximally entangled states in C ^{<em>d</em>₁} ⊗ (C ^{<em>d</em>₂} ) ^{⊗2<em>n</em>} .<br></p>