Entropic characterization of stabilizer states and magic states

Quantum states with minimum or maximum uncertainty are of special significance due to their extreme properties. Celebrated examples are coherent states induced from certain Lie groups and intelligent states for various uncertainty relations. In this work, by virtue of the Maassen-Uffink entropic uncertainty relation, we introduce an entropic quantifier of uncertainty and use it to characterize several important families of states in the stabilizer formalism of quantum computation. More specifically, we show that the stabilizer states and T-type magic states stand at the two extremes of the entropic quantifier of uncertainty: The former are precisely the minimum entropic uncertainty states, while the latter are precisely the maximum entropic uncertainty states. Moreover, interpolating between the above two extremes, the H-type magic states are the saddle points of the entropic quantifier of uncertainty. These entropic characterizations reveal some intrinsic features of stabilizer states, H- and T-type magic states, and cast novel light on the resource-theoretic viewpoint of regarding the stabilizer states as free states and the T-type magic states as the most precious source states in the stabilizer quantum theory.