The Collatz-Wielandt quotient for pairs of nonnegative operators

TL;DR

This paper explores the Collatz-Wielandt quotient for pairs of nonnegative operators, providing characterizations and bounds relevant to applications like commodity pricing and quantum information.

Contribution

It introduces a comprehensive analysis of the Collatz-Wielandt quotient for pairs of nonnegative operators, including rectangular matrices and completely positive operators, with computable bounds.

Findings

Characterization of minimal optimal solutions.

Polynomially computable bounds on the quotient.

Application relevance to various fields such as quantum information.

Abstract

In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators A,B that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and B is the identity operator then one version of this quotient is the spectral radius of A. In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with the Collatz-Wielandt quotient for two nonnegative operators. In this paper we treat the two important cases: a pair of rectangular nonnegative matrices and a pair completely positive operators. We give a characterization of minimal optimal solutions and polynomially computable bounds on the Collatz-Wielandt quotient.