While positive linear maps send positive semidefinite (PSD) matrices to PSD matrices, they may introduce negative eigenvalues when only applied to a subsystem---this is precisely the property that makes them useful for entanglement detection in quantum information theory.

In this talk, we investigate the question of exactly how negative those eigenvalues can be, and more generally what the possible spectra of $(I \otimes \Phi)(X)$ are when $\Phi$ is a positive map and $X$ is a PSD matrix.