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

Abstract: 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, we define a "MUB-balanced state" to be a pure state for which the list of probabilities of the $d$ outcomes of one of these measurements is independent of the choice of measurement, up to permutations. In this paper we explicitly construct a MUB-balanced state for each prime power dimension $d$ for which $d = 3$ (mod 4). These states have already been constructed by Appleby in unpublished notes, but our presentation here is different in that both the expression for the states themselves and the proof of MUB-balancedness are given in terms of the discrete Wigner function, rather than the density matrix or state vector. The discrete Wigner functions of these states are "rotationally symmetric" in a sense roughly analogous to the rotational symmetry of the energy eigenstates of a harmonic oscillator in the continuous two-dimensional phase space. Upon converting the Wigner function to a density matrix, we find that the states are expressible as real state vectors in the standard basis. We observe numerically that when $d$ is large (and not a power of 3), a histogram of the components of such a state vector appears to form a semicircular distribution.