Completely positive semidefinite rank

  title={Completely positive semidefinite rank},
  author={Anupam Prakash and Jamie Sikora and Antonios Varvitsiotis and Zhaohui Wei},
  journal={Mathematical Programming},
An $$n\times n$$n×n matrix X is called completely positive semidefinite (cpsd) if there exist $$d\times d$$d×d Hermitian positive semidefinite matrices $$\{P_i\}_{i=1}^n$${Pi}i=1n (for some $$d\ge 1$$d≥1) such that $$X_{ij}= \mathrm {Tr}(P_iP_j),$$Xij=Tr(PiPj), for all $$i,j \in \{ 1, \ldots , n \}$$i,j∈{1,…,n}. The cpsd-rank of a cpsd matrix is the smallest $$d\ge 1$$d≥1 for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate in two… 
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