Universal Bound on the Eigenvalues of 2-Positive Trace-Preserving Maps

TL;DR

This paper establishes a tight upper bound on the trace of 2-positive, trace-preserving maps based on their smallest eigenvalue, generalizing known results with a simpler algebraic proof.

Contribution

It introduces a universal spectral bound for 2-positive trace-preserving maps and extends it to generators of semigroups, simplifying previous proofs.

Findings

Bound is tight and necessary for 2-positivity

Extension to generators of semigroups

Simplified algebraic proof of spectral bounds

Abstract

We prove an upper bound on the trace of any 2-positive, trace-preserving map in terms of its smallest eigenvalue. We show that this spectral bound is tight, and that 2-positivity is necessary for this inequality to hold in general. Moreover, we use this to infer a similar bound for generators of one-parameter semigroups of 2-positive trace-preserving maps. With this approach we generalize known results for completely positive trace-preserving dynamics while providing a significantly simpler proof that is entirely algebraic.