Mutually Unbiased Product Bases

A pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are complementary, i.e. a measurement of one observable implies maximal uncertainty about the possible outcome of a subsequent measurement of a second observable. MU bases have attracted interest in recent years because their properties seem to depend dramatically on the dimension d of the quantum system in hand. If the dimension is given by a prime or prime-power, the state space is known to accommodate a complete set of d+1 MU bases. However, for composite dimensions, such as d = 6, 10, 12, ..., complete sets seem to be absent and it is not understood why. In this thesis we carry out a comprehensive study of MU product bases in dimension six. In particular, we construct all MU bases in dimension six consisting of product states only. The exhaustive classification leads to several non-existence results. We also present a new construction of complex Hadamard matrices of composite order, which is a consequence of our work on MU product bases. Based on this construction we obtain several new isolated Hadamard matrices of Butson-type.