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

### Abstract

A complete set of mutually unbiased bases (MUBs) 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 any 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 These states have already been constructed by Appleby in unpublished notes, but our presentation here is different in that both the expression for the states themselves and the proof of MUB-balancedness are given in terms of the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.

© 2014 AIP Publishing LLC

Received 26 July 2014
Accepted 04 December 2014
Published online 23 December 2014

Acknowledgments:
We are grateful for discussions and email correspondence with Marcus Appleby. We also thank Steven Miller, Ron Evans, and Nick Katz for their interest in the semicircular distribution suggested by our numerical results and for pursuing an explanation. Research by W.K.W. is supported in part by the Foundational Questions Institute (Grant No. FQXi-RFP3-1350).

Article outline:

I. INTRODUCTION
II. THE DISCRETE WIGNER FUNCTION
III. PROOF THAT *W*
_{
ρ
} IS THE WIGNER FUNCTION OF A PURE STATE
IV. PROOF THAT THE STATE *ρ* IS MUB-BALANCED
V. THE DENSITY MATRIX AND THE STATE VECTOR
VI. CONCLUSIONS

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2014-12-23

2016-09-26

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