2006/11/30 by D. Gross, Katrien Audenaert, K. Audenaert +2 · 7 citations
Decision Sciences · Engineering · Materials Science · Mathematics · Physics and Astronomy · #Algebra over a field #Algorithm #Cardinality (data modeling) #Character (mathematics) #Circular ensemble #Combinatorics #Computer science #Discrete mathematics #Geometry #Mathematical analysis #Mathematics #Mutually unbiased bases #Optimal Experimental Design Methods #Order (exchange) #Polynomial #Pure mathematics #Quasicrystal Structures and Properties #Set (abstract data type) #Simple (philosophy) #Sketch #Unitary group #Unitary matrix #Unitary state #graph theory and CDMA systems #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1063/1.2716992
published as J. Math. Phys. 48, 052104 (2007) · 15 pages, one figure. Minor revisions to mirror version to appear in J. Math. Phys
openalex publication_date 2007/05/01 · arxiv created 2007/05/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 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.