Let v1,..., vn be n vectors in an inner product space. Can we find a
natural number d and positive (semidefinite) complex matrices A1,...,
An of size d×d such that Tr(AkAl)= for all
k,l=1,...,n? For such matrices to exist, one must have ≥0
for all k,l=1,...,n. We prove that if n<5 then this trivial necessary
condition is also a sufficient one and find an appropriate example showing that
from n=5 this is not so --- even if we allowed realizations by positive
operators in a von Neumann algebra with a faithful normal tracial state.
The fact that the first such example occurs at n=5 is similar to what one
has in the well-investigated problem of positive factorization of positive
(semidefinite) matrices. If the matrix () has a positive
factorization, then matrices A1,..., An as above exist. However, as we
show by a large class of examples constructed with the help of the Clifford
algebra, the converse implication is false.Comment: 8 page