Spectral majorization under a unital quantum channel ($\lambda (\Lambda (\rho ))\prec \lambda (\rho )$)

A unital quantum channel mixes its input spectrum with a doubly stochastic matrix, so its output spectrum satisfies the majorization relation with its input spectrum. The reverse inequality is generally false: a pure state can be sent to a maximally mixed state. The direction expresses the fact that unital noise can make eigenvalues more uniform. Equality of the two spectra is possible, for example under a unitary channel.