Uncertainty characterization of stabilizer states and magic states for qubit systems

For qubit systems, we propose three quantifiers of uncertainty given in the product form of uncertainties for Pauli observables. We analyze the corresponding quantum states which achieve the minimum and maximum of these quantifiers, and reveal their connections with the stabilizer formalism of fault-tolerant quantum computation. Explicitly, for the quantifier of total and quantum uncertainty, we show that the minimum and maximum uncertainty states are the stabilizer states and the T-type magic states, respectively. We compare our results with two recently proposed characterizations of stabilizer states and magic states via the Heisenberg uncertainty relations and refined quantum uncertainty relations. Also, we briefly discuss the situation when the uncertainty quantifiers are defined in the sum form.