2009/12/31 by D. M. Appleby, Steven T. Flammia, Christopher A. Fuchs · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebra over a field #Algebraic number #Algebraic structure #Basis (linear algebra) #Dimension (graph theory) #Lie algebra #Quantum Information and Cryptography #Quantum Mechanics and Applications #Representation theory #Structure constants #Triple system #Unitary state #math-ph #math.CO #math.MP #quant-ph
paper · pdf · doi:10.1063/1.3555805
published as J. Math. Phys. 52, 022202 (2011) · 56 pages
arxiv created 2009/12/31 · openalex publication_date 2011/02/01 · arxiv updated 2011/02/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 \documentclass[12pt]minimal\begindocumentgl(d,\mathbb C)\enddocument 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 \documentclass[12pt]minimal\begindocumentgl(d,\mathbb C)\enddocument 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.