Physics – Quantum Physics
Scientific paper
2006-10-31
J. Math. Phys. 48, 052104 (2007)
Physics
Quantum Physics
15 pages, one figure. Minor revisions to mirror version to appear in J. Math. Phys
Scientific paper
10.1063/1.2716992
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.
Audenaert Koenraad
Eisert Jens
Gross David
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