Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A pure quantum state is called $k$-uniform if all its reductions to $k$-qudit are maximally mixed.  investigate the general constructions of $k$-uniform pure quantum states of $n$ subsystems with $d$ levels.  provide one construction via symmetric matrices and the second one through classical error-correcting codes. There are three main results arising from our constructions. Firstly,  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 $p$. Secondly, both constructions show that their 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 outperforms the one by error-correcting codes. Thirdly, our symmetric matrix construction provides a positive ansr to the open question in \cite{DA} on whether there exists $3$-uniform $n$-qudit pure quantum state for all $n\ge 8$. In fact,  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 concatenation of algebraic geometry codes,  give an explicit construction of $k$-uniform quantum state when $k$ tends to infinity.
['Keqin Feng', 'Lingfei Jin', 'Chaoping Xing', 'Chen Yuan']
I am writing a research paper on a idea Multipartite entangled states, symmetric matrices and error-correcting codes
[{"from": "human", "value": "I am writing a research paper on a idea Multipartite entangled states, symmetric matrices and error-correcting codes"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A pure quantum state is called $k$-uniform if all its reductions to $k$-qudit are maximally mixed.  investigate the general constructions of $k$-uniform pure quantum states of $n$ subsystems with $d$ levels.  provide one construction via symmetric matrices and the second one through classical error-correcting codes. There are three main results arising from our constructions. Firstly,  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 $p$. Secondly, both constructions show that their 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 outperforms the one by error-correcting codes. Thirdly, our symmetric matrix construction provides a positive ansr to the open question in \\cite{DA} on whether there exists $3$-uniform $n$-qudit pure quantum state for all $n\\ge 8$. In fact,  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 concatenation of algebraic geometry codes,  give an explicit construction of $k$-uniform quantum state when $k$ tends to infinity."}]