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I introduce a new notion, that extends the mutually unbiased bases (MUB)conditons to more than two bases. These, I call the nUB conditions, and thecorresponding bases $n$-fold unbiased. They naturally appear while optimizinggeneric $n$-to-one quantum random access code (QRAC) strategies. While theirexistence in general dimensions is an open question, they nevertheless giveclose-to-tight upper bounds on QRAC success probabilities, and raisefundamental questions about the geometry of quantum states.

$n$-fold unbiased bases: an extension of the MUB condition