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The Lie algebraic significance of symmetric informationally complete measurements

Appleby, D. M. and Flammia, Steven T. and Fuchs, Christopher A. (2011) The Lie algebraic significance of symmetric informationally complete measurements. Journal of Mathematical Physics, 52 (2). Art. No. 022202. ISSN 0022-2488.

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Examples of symmetric informationally complete positive operator-valued measures (SIC-POVMs) have been constructed in every dimension ⩽67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl (d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl (d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.

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Additional Information:© 2011 American Institute of Physics. Received 15 February 2010; accepted 26 January 2011; published online 25 February 2011. The authors thank I. Bengtsson for discussions. The work of D.M.A. and C.A.F. was supported in part by the U. S. Office of Naval Research (Grant No. N00014-09-1-0247). Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
Funding AgencyGrant Number
U.S. Office of Naval ResearchN00014-09-1-0247
Government of Canada through Industry CanadaUNSPECIFIED
Province of Ontario through the Ministry of Research and InnovationsUNSPECIFIED
Subject Keywords:Lie algebras, quantum crytopography, quantum theory
Issue or Number:2
Classification Code:PACS: 03.65.Fd; 03.65.Ta; 03.65.Wj; 03.67.Dd; 02.10.Ud
Record Number:CaltechAUTHORS:20110318-145128936
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Official Citation:The Lie algebraic significance of symmetric informationally complete measurements D. M. Appleby, Steven T. Flammia, and Christopher A. Fuchs J. Math. Phys. 52, 022202 (2011); doi:10.1063/1.3555805
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:22982
Deposited By: Ruth Sustaita
Deposited On:21 Mar 2011 20:15
Last Modified:03 Oct 2019 02:42

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