States that are far from being stabilizer states

Abstract: Stabilizer states are eigenvectors of maximal commuting sets of operators in a finite Heisenberg group. States that are far from being stabilizer states include magic states in quantum computation, MUB-balanced states, and SIC vectors. In prime dimensions the latter two fall under the umbrella of Minimum Uncertainty States (MUS) in the sense of Wootters and Sussman. We study the correlation between two ways in which the notion of "far from being a stabilizer state" can be quantified, and give detailed results for low dimensions. In dimension 7 we identify the MUB-balanced states as being antipodal to the SIC vectors within the set of MUS, in a sense that we make definite. In dimension 4 we show that the states that come closest to being MUS with respect to all the six stabilizer MUBs are the fiducial vectors for Alltop MUBs.