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Discrete Phase-Space Structure and Mutually Unbiased Basis Operators

Several construction methods for mutually unbiased bases have been proposed in the literature.  Typically they involve either direct construction of the basis vectors, or sets of operators are derived where each set's (simultaneous) eigenstates are mutually unbiased with respect to every other set's.  It is subsequently common to map the states onto lines in the corresponding discrete phase space.  We show how to derive mutually unbiased bases from the reverse mapping.  We start by considering the most general phase-space structures compatible with the concept of mutually unbiased bases, namely bundles of discrete space curves intersecting only at the origin and satisfying certain properties and develop a new method based on the analysis of geometrical structures in the finite phase-space for construction of Mutually Unbiased Bases (MUB) operators.  In the case when the Hilbert space dimension is an integer power of a prime, there exist several classes of curve bundles with different properties, lines being a special case.  We also consider transformations between different kinds of curves, and show that in the two-qubit case, they all correspond to local transformations, and more specifically they correspond to rotations around the Bloch-sphere principal axes.  Nevertheless, in the case of more that two qubits several non isomorphic structures appear, which can be naturally classified in terms of discrete curve bundles.

The existence and a possibility of regular searching of such non isomorphic MUBs allows us to introduce a concept of complexity of tomographic scheme for state determination of multi-qubit systems.