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Physics Letters A

Constructing symmetric informationally complete positive-operator-valued measures in Bloch space

Roberto Salazar, Dardo Goyeneche, A. Delgado, C. Saavedra

Published Dec 13, 2011• 8 citations
Publisher
DOI
10.1016/j.physleta.2011.10.074

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Abstract In quantum information theory, symmetric informationally complete positive operator-valued measures (SIC-POVMs) are related to quantum state tomography, quantum cryptography and foundational studies. It is difficult to construct SIC-POVMs and remains unknown whether SIC-POVMs exist for infinitely dimension. Therefore, researchers have proposed the solution to construct approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs). This paper proposes a construction of ASIC-POVMs using character sums over Galois rings. The dimension of this ASIC-POVMs is q-1, where q is a prime power. Also, our constructions include partial results about ASIC-POVMs constructed by X. Cao et al (X. Cao, J. Mi and S. Xu: Two constructions of approximately symmetric informationally complete positive operator-valued measures. J. Math. Phys., 58(6), 062201 (2017)). Mathematics Subject Classification (2020) 81P15 · 81P45
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension d^2-1 corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC-POVMs contains weak SIC-POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC-POVM.