Gross, D., Audenaert, K. and Eisert, J. (2007) Evenly distributed unitaries: on the structure of unitary designs. J. Math. Phys., 48
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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 $t$-th 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 whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.
This is a Submitted version This version's date is: 2007 This item is not peer reviewed
https://repository.royalholloway.ac.uk/items/e95a7bbf-708d-e8c4-28ca-648237ebb281/1/
Deposited by Research Information System (atira) on 23-May-2012 in Royal Holloway Research Online.Last modified on 23-May-2012
15 pages, one figure. Minor revisions to mirror version to appear in J. Math. Phys