Evenly distributed unitaries: On the structure of unitary designs

@article{Gross2007EvenlyDU,
  title={Evenly distributed unitaries: On the structure of unitary designs},
  author={David Gross and Koenraad M. R. Audenaert and Jens Eisert},
  journal={Journal of Mathematical Physics},
  year={2007},
  volume={48},
  pages={052104-052104}
}
We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs… 
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