# Exploring the bounds on the positive semidefinite rank

## TL;DR

This paper investigates the limitations of existing bounds on the positive semidefinite rank of matrices related to polytopes, showing they cannot produce exponential lower bounds on extension complexity, and relates these bounds to the matrix's regular rank.

## Contribution

It proves that current bounds on PSD-rank are polynomially bounded by the regular rank, providing new insights into extension complexity limitations.

## Key findings

- Existing bounds are upper bounded by polynomial functions of regular rank.
- No exponential lower bounds on PSD-rank can be derived from current bounds.
- An upper bound on mutual information based on regular rank is established.

## Abstract

The nonnegative and positive semidefinite (PSD-) ranks are closely connected to the nonnegative and positive semidefinite extension complexities of a polytope, which are the minimal dimensions of linear and SDP programs which represent this polytope. Though some exponential lower bounds on the nonnegative and PSD- ranks has recently been proved for the slack matrices of some particular polytopes, there are still no tight bounds for these quantities. We explore some existing bounds on the PSD-rank and prove that they cannot give exponential lower bounds on the extension complexity. Our approach consists in proving that the existing bounds are upper bounded by the polynomials of the regular rank of the matrix, which is equal to the dimension of the polytope (up to an additive constant). As one of the implications, we also retrieve an upper bound on the mutual information of an arbitrary matrix of a joint distribution, based on its regular rank.