Discrete phase-space structure of n-qubit mutually unbiased bases

We work out the phase-space structure for a system of n qubits. We replace the field of real numbers that label the axes of the continuous phase space by the finite field GF(2ⁿ) and investigate the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We provide a simple classification of such curves and study in detail the four- and eight-dimensional cases, analyzing also the effect of local transformations. In this way, we provide a comprehensive phase-space approach to the construction of mutually unbiased bases for n qubits.