×

zbMATH — the first resource for mathematics

Evenly distributed unitaries: on the structure of unitary designs. (English) Zbl 1144.81351
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.

MSC:
81P40 Quantum coherence, entanglement, quantum correlations
05B30 Other designs, configurations
81V80 Quantum optics
94A20 Sampling theory in information and communication theory
22E30 Analysis on real and complex Lie groups
43A80 Analysis on other specific Lie groups
47N50 Applications of operator theory in the physical sciences
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] G. Zauner, Ph.D. thesis, University of Vienna, 1999;
[2] DOI: 10.1063/1.1896384 · Zbl 1110.81023
[3] DOI: 10.1063/1.1896384 · Zbl 1110.81023
[4] DOI: 10.1103/PhysRevA.72.032325
[5] Klappenecker A., Proceedings of the IEEE International Symposium on Information Theory pp 1740– (2005)
[6] DOI: 10.1088/0305-4470/39/43/009 · Zbl 1107.81017
[7] DOI: 10.1103/PhysRevA.65.044301
[8] DOI: 10.1103/PhysRevA.65.044301
[9] DOI: 10.1103/PhysRevA.65.044301
[10] C. Dankert M.Sc. thesis, University of Waterloo, 2005; also available at e-print quant-ph/0512217.
[11] Fuchs C. A., Quantum Inf. Comput. 3 pp 377– (2003)
[12] DOI: 10.1109/TIT.2005.844076 · Zbl 1294.94091
[13] DOI: 10.1109/18.985948 · Zbl 1071.81511
[14] DOI: 10.1103/PhysRevA.40.4277 · Zbl 1371.81145
[15] DOI: 10.1103/PhysRevA.61.062313
[16] DOI: 10.1103/PhysRevA.72.052326
[17] Issacs I. M., Character Theory of Finite Groups (1976)
[18] Breuer T., Manual for the GAP Character Table Library Version 1.1 (2004)
[19] DOI: 10.1088/1367-2630/7/1/073
[20] DOI: 10.1088/1367-2630/7/1/073
[21] DOI: 10.1063/1.2393152 · Zbl 1112.81012
[22] D. Gross, thesis, University of Potsdam, 2005;
[23] Jungnickel D., Finite fields (1993)
[24] DOI: 10.1088/0305-4470/38/26/012 · Zbl 1073.81058
[25] DOI: 10.1103/PhysRevA.71.062310
[26] DOI: 10.1007/978-3-642-64981-3 · Zbl 0217.07201
[27] Simon B., Representations of Finite and Compact Groups (1996) · Zbl 0840.22001
[28] Fulton W., Representation Theory (1991)
[29] DOI: 10.1103/RevModPhys.79.135
[30] DOI: 10.1007/BF02848172
[31] DOI: 10.1142/S0219749903000371 · Zbl 1069.81508
[32] DOI: 10.1103/PhysRevLett.90.047904
[33] DOI: 10.1103/PhysRevLett.72.1148 · Zbl 0973.81502
[34] DOI: 10.1103/PhysRevLett.72.1148 · Zbl 0973.81502
[35] DOI: 10.1103/PhysRevLett.72.1148 · Zbl 0973.81502
[36] DOI: 10.1007/s00220-006-1535-6 · Zbl 1107.81011
[37] Dahlsten O., Stat. Probab. Lett. 6 pp 527– (2006)
[38] DOI: 10.1088/0305-4470/34/35/335 · Zbl 1031.81011
[39] DOI: 10.1016/j.jsc.2005.01.002 · Zbl 1124.20007
[40] DOI: 10.1023/A:1021323312367 · Zbl 1028.42022
[41] DOI: 10.1016/0001-8708(84)90022-7 · Zbl 0596.05012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.