The geometry of mutually unbiased basis

Mutually unbiased bases, often called MUBs, are highly constrained geometric objects in an n dimensional complex vector space, Cn. MUBs are often studied for their applications in quantum physics and related fields, including quantum cryptography. The volatile nature of particles in quantum physics, when combined with MUBs, can be used to create useful cryptographic protocols. MUBs are interesting as they are highly constrained geometric objects over a continuous field, but have strong links to discreet structures like finite fields, mutually orthogonal latin squares and lattices. There are open problems relating MUBs to discrete structures and the maximal possible set size of MUBs. Taking into consideration the geometry of MUBs, the links to discrete structures, and the open problems for MUBs, this thesis highlights the underlying geometry of MUBs. The thesis achieves this by developing the use of lattices and lattice walks as techniques for examining MUBs, developing a new construction for MUBs using groups, and proving that several types of MUBs containing Butson-Hadamard matrices cannot exist by utilising a new necessary condition for the existence of MUBs containing Butson- Hadamard matrices. In Chapter 1, a historical background for MUBs and their applications provides motivation for research on MUBs, the research objectives for the thesis are presented, a discussion of the scope of the research in the thesis is included, and a summary of the thesis is provided. In Chapter 2, a review of the literature, definitions, theorems, construction methods, open problems, conjectures, a discussion of the properties of MUBs, and a summary of the research approach taken in this thesis are presented. 1 Summary In Chapter 3, lattices are developed as part of a technique to examine MUBs. By partially calculating the Hermitian dot product between two unbiased vectors, you get a sum of complex numbers that can be represented as a walk in the complex plane. If the types of vectors used are restricted to contain only complex roots of unity, these walks can be further restricted to special lattices that are a generalisation of J-lattices. These lattices that we call m-complex lattices are interesting in their own right, as they are often topologically dense over C. Once these lattices are established, the concept of an unbiased lattice walk forms and can be used as a necessary condition for the existence of MUBs containing Butson-Hadamard matrices. Using a simple algorithm, a computation search is performed on small orders of unbiased walks to prove when unbiased lattice walks exist and do not exist. In Chapter 4 several smaller research results related to the geometry of MUBs and construction methods for MUBs are presented. There are several construction methods for MUBs, and many of these constructions use discrete objects including finite fields, complex roots of unity, unimodular complex Hadamard matrices, planar functions, and Alltop functions. By deconstructing these constructions, links with groups emerge, and the examination of these groups combined with linear algebra as a tool allows the development of a new generalised group construction method for MUBs. These groups and linear algebra also help to examine the potential equivalence of certain sets of MUBs. A thought experiment is explored with 1 dimensional MUBs to help highlight how the orthonormal property restricts the set size in MUBs. In Chapter 5 a conclusion summarising the results and several new conjectures is presented, highlighting potential avenues for further research. Appendix A contains a list of lattice walks found by the algorithm in Chapter 3. Appendix B contains 2 version of the code used to find lattice walks, as discussed in Chapter 3.