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A complete set of mutually unbiased bases in a Hilbert space of dimension $d$defines a set of $d+1$ orthogonal measurements. Relative to such a set, wedefine a "MUB-balanced state" to be a pure state for which the list ofprobabilities of the $d$ outcomes of one of these measurements is independentof the choice of measurement, up to permutations. In this paper we explicitlyconstruct a MUB-balanced state for each prime power dimension $d$ for which $d= 3$ (mod 4). These states have already been constructed by Appleby inunpublished notes, but our presentation here is different in that both theexpression for the states themselves and the proof of MUB-balancedness aregiven in terms of the discrete Wigner function, rather than the density matrixor state vector. The discrete Wigner functions of these states are"rotationally symmetric" in a sense roughly analogous to the rotationalsymmetry of the energy eigenstates of a harmonic oscillator in the continuoustwo-dimensional phase space. Upon converting the Wigner function to a densitymatrix, we find that the states are expressible as real state vectors in thestandard basis. We observe numerically that when $d$ is large (and not a powerof 3), a histogram of the components of such a state vector appears to form asemicircular distribution.

States that "look the same" with respect to every basis in a mutually unbiased set