Rigidity and a common framework for mutually unbiased bases and k ‐nets

Many deep connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called k ‐nets (and in particular, between collections of MUBs and finite affine planes). Here we introduce the notion of a k ‐net over a C * ‐algebra, providing a common framework for both objects. In the commutative case, we recover (classical) k ‐nets, while the choice of M d ( C ) leads to collections of MUBs. In this framework, we derive a rigidity property which hence automatically applies to both objects. For k ‐nets that can be completed to affine planes, this was already known by a completely different, combinatorial argument. For k ‐nets that cannot be completed and for MUBs, this result is new, and it implies that the only vectors unbiased to all but k ≤ d bases of a complete collection of MUBs in C d are the elements of the remaining k bases (up to phase factors). Further, we show that this bound is tight with counterexamples for k > d in every prime‐square dimension. Applying our rigidity result, we prove that if a large enough collection of MUBs constructed from a certain unitary error basis (like, the generalized Pauli operators) can be extended to a complete system, then every basis of the completion must come from the same error basis. In turn, we use this to show that certain large systems of MUBs cannot be completed.