On vector configurations that can be realized in the cone of positive matrices

TL;DR

This paper investigates when vectors in an inner product space can be realized as traces of products of positive semidefinite matrices, establishing that for fewer than five vectors the positive inner product condition is sufficient, but not for five or more.

Contribution

The authors prove that the positive inner product condition is sufficient for vector realization with positive matrices when fewer than five vectors, and provide counterexamples for five or more.

Findings

For n<5, positive inner product matrices are realizable with positive matrices.

Counterexamples exist for n≥5 showing realizability fails despite positive inner products.

Construction of examples using Clifford algebra demonstrates the limitations of positive factorizations.

Abstract

Let $v_1$,..., $v_n$ be $n$ vectors in an inner product space. Can we find a natural number $d$ and positive (semidefinite) complex matrices $A_1$,..., $A_n$ of size $d \times d$ such that ${\rm Tr}(A_kA_l)= <v_k, v_l>$ for all $k,l=1,..., n$? For such matrices to exist, one must have $<v_k, v_l> \geq 0$ 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 $(<v_k, v_l>)$ has a positive factorization, then matrices $A_1$,..., $A_n$ 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.