Characterization of brain states is essential for understanding its functioning in the absence of external stimuli. Brain states differ on their balance between excitation and inhibition, and on the diversity of their activity patterns. These can be respectively indexed by 1/f slope and Lempel-Ziv complexity (LZc). However, whether and how these two brain state properties relate remain elusive. Here we analyzed the relation between 1/f slope and LZc with two in-silico approaches and in both rat EEG and monkey ECoG data. We contrasted resting state with propofol anesthesia, which directly modulates the excitation-inhibition balance. We found convergent results among simulated and empirical data, showing a strong, inverse and non trivial monotonic relation between 1/f slope and complexity, consistent at both ECoG and EEG scales. We hypothesize that differentially entropic regimes could underlie the link between the excitation-inhibition balance and the vastness of the repertoire of brain systems.