Mutually unbiased bases: The standard construction and automorphisms

A set of mutually unbiased bases is a set of orthonormal bases of a complex vector space for which a certain inner-product condition holds for pairs of vectors from distinct bases. There is a known bound on the maximum possible cardinality of a set of mutually unbiased bases; sets which achieve this maximal cardinality are known as complete sets. Most of the results in this dissertation concern complete sets.;In particular, we establish two distinct, new proofs of the fact that a complete set of mutually unbiased bases of order 5 is essentially unique. Our proofs borrow techniques used in the study of finite geometry. We also make use of cyclic n-roots and Hilbert polynomials in these proofs. In addition to these two new proofs, we provide a result which states that under certain restrictive conditions, a complete set of odd prime order must be essentially unique.;We define automorphisms of mutually unbiased bases. We establish the full automorphism group of the so-called standard construction of a complete set of mutually unbiased bases in odd prime cases, again using tools from finite geometry. We further establish a result which restricts the form complete sets can take based on their automorphism groups.