Completely positive semidefinite rank

Anupam Prakash, Jamie Sikora, Antonios Varvitsiotis, Zhaohui Wei

Abstract: 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 $$\{P_i\}_{i=1}^n$$ { P i } i = 1 n (for some $$d\ge 1$$ d ≥ 1 ) such that $$X_{ij}= \mathrm {Tr}(P_iP_j),$$ X i j = Tr ( P i P j ) , 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 ways. First, the cpsd-rank is a natural non-commutative analogue of the completely positive rank of a completely positive matrix. Second, we show that the cpsd-rank is physically motivated as it can be used to upper and lower bound the size of a quantum system needed to generate a quantum behavior. In this work we present several properties of the cpsd-rank. Unlike the completely positive rank which is at most quadratic in the size of the matrix, no general upper bound is known on the cpsd-rank of a cpsd matrix. In fact, we show that the cpsd-rank can be sub-exponential in terms of the size. Specifically, for any $$n\ge 1,$$ n ≥ 1 , we construct a cpsd matrix of size 2n whose cpsd-rank is $$2^{\varOmega (\sqrt{n})}$$ 2 Ω ( n ) . Our construction is based on Gram matrices of Lorentz cone vectors, which we show are cpsd. The proof relies crucially on the connection between the cpsd-rank and quantum behaviors. In particular, we use a known lower bound on the size of matrix representations of extremal quantum correlations which we apply to high-rank extreme points of the n-dimensional elliptope. Lastly, we study cpsd-graphs, i.e., graphs G with the property that every doubly nonnegative matrix whose support is given by G is cpsd. We show that a graph is cpsd if and only if it has no odd cycle of length at least 5 as a subgraph. This coincides with the characterization of cp-graphs.