Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $A$ be a matrix with nonnegative real entries. The PSD rank of $A$ is the smallest integer $k$ for which there exist $k\times k$ real PSD matrices $B_1,\ldots,B_m$, $C_1,\ldots,C_n$ satisfying $A(i|j)=\operatorname{tr}(B_iC_j)$ for all $i,j$.  determines the computational complexity status of the PSD rank. Namely,  show that the problem of computing this function is polynomial-time equivalent to the existential theory of the reals.
['Yaroslav Shitov']
I am writing a research paper on a idea The complexity of positive semidefinite matrix factorization
[{"from": "human", "value": "I am writing a research paper on a idea The complexity of positive semidefinite matrix factorization"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $A$ be a matrix with nonnegative real entries. The PSD rank of $A$ is the smallest integer $k$ for which there exist $k\\times k$ real PSD matrices $B_1,\\ldots,B_m$, $C_1,\\ldots,C_n$ satisfying $A(i|j)=\\operatorname{tr}(B_iC_j)$ for all $i,j$.  determines the computational complexity status of the PSD rank. Namely,  show that the problem of computing this function is polynomial-time equivalent to the existential theory of the reals."}]