2008/02/06 by Aleksandrs Belovs, Belovs, Aleksandrs, Juris Smotrovs +1
Computer Science · Engineering · Physics and Astronomy · #Coding theory and cryptography #FOS: Physical sciences #Quantum Physics (quant-ph) #graph theory and CDMA systems #quant-ph #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0802.0855
19 pages: revised and heavily extended in the part concerning MUBs
openalex publication_date 2008/02/06 · arxiv created 2008/07/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper gives a short introduction to mutually unbiased bases and the Welch bounds and demonstrates that the latter is a good technical tool to explore the former. In particular, a criterion for a system of vectors to satisfy the Welch bounds with equality is given and applied for the case of MUBs. This yields a necessary and sufficient condition on a set of orthonormal bases to form a complete system of MUBs. This condition takes an especially elegant form in the case of homogeneous systems of MUBs. We express some known constructions of MUBs in this form. Also it is shown how recently obtained results binding MUBs and some combinatorial structures (such as perfect nonlinear functions and relative difference sets) naturally follow from this criterion. Some directions for proving non-existence results are sketched as well.