Rational and real positive semidefinite rank can be different

TL;DR

This paper demonstrates that for certain nonnegative matrices with rational entries, the rational-restricted psd rank can be strictly greater than the usual psd rank, highlighting a fundamental difference between these two measures.

Contribution

It provides the first example showing that rational-restricted psd rank can be strictly larger than the usual psd rank for a nonnegative matrix.

Findings

Rational-restricted psd rank can exceed the usual psd rank.

An explicit example of a matrix with psd rank four and higher rational-restricted psd rank.

The inequality between these ranks can be strict, contrary to initial expectations.

Abstract

Given a nonnegative matrix M with rational entries, we consider two quantities: the usual positive semidefinite (psd) rank, where the matrix is factored through the cone of real symmetric psd matrices, and the rational-restricted psd rank, where the matrix factors are required to be rational symmetric psd matrices. It is clear that the rational-restricted psd rank is always an upper bound to the usual psd rank. We show that this inequality may be strict by exhibiting a matrix with psd rank four whose rational-restricted psd rank is strictly greater than four.