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

Nietert, Sloan and Szilágyi, Zsombor and Weiner, Mihály (2020) Rigidity and a common framework for mutually unbiased bases and k-nets. JOURNAL OF COMBINATORIAL DESIGNS, 28 (12). pp. 869-892. ISSN 1063-8539

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Abstract

Many deep connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs calledk-nets (and in particular, between collections of MUBs and finite affine planes). Here we introduce the notion of ak-net over aC*-algebra, providing a common framework for both objects. In the commutative case, we recover (classical)k-nets, while the choice ofMd(C)leads to collections of MUBs. In this framework, we derive a rigidity property which hence automatically applies to both objects. Fork-nets that can be completed to affine planes, this was already known by a completely different, combinatorial argument. Fork-nets that cannot be completed and for MUBs, this result is new, and it implies that the only vectors unbiased to all butk <= dbases of a complete collection of MUBs inCdare the elements of the remainingkbases (up to phase factors). Further, we show that this bound is tight with counterexamples fork>din 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, theneverybasis 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.

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