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

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.