Multipartite entangled states, symmetric matrices and error-correcting codes

Abstract

A pure quantum state is called $k$ -uniform if all its reductions to $k$ -qudit are maximally mixed. We investigate the general constructions of $k$ -uniform pure quantum states of $n$ subsystems with $d$ levels. We provide one construction via symmetric matrices and the second one through the classical error-correcting codes. There are three main results arising from our constructions. First, we show that for any given even $n\ge 2$ , there always exists an $n/2$ -uniform $n$ -qudit quantum state of level $p$ for sufficiently large prime $p$ . Second, both constructions show that there exist $k$ -uniform $n$ -qudit pure quantum states such that $k$ is proportional to $n$ , i.e., $k=\Omega (n)$ although the construction from symmetric matrices in general outperforms the one by error-correcting codes. Third, our symmetric matrix construction provides a positive answer to the open question on whether there exists a 3-uniform $n$ -qudit pure quantum state for all $n\ge 8$ . In fact, we can further prove that, for every $k$ , there exists a constant $M_{k}$ such that there exists a $k$ -uniform $n$ -qudit quantum state for all $n\ge M_{k}$ . In addition, by using the concatenation of algebraic geometry codes, we give an explicit construction of $k$ -uniform quantum state when $k$ tends to infinity.