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A set of mutually unbiased bases (MUBs) is said to be unextendible if theredoes not exist another basis that is unbiased with respect to the given set.Here, we prove the existence of smaller sets of MUBs in prime-squareddimensions ($d=p^{2}$) that cannot be extended to a complete set using thegeneralized Pauli operators. We further observe an interesting connectionbetween the existence of unextendible sets and the tightness of entropicuncertainty relations (EURs) in these dimensions. In particular, we show thatour construction of unextendible sets of MUBs naturally leads to sets of $p+1$MUBs that saturate both a Shannon ($H_{1}$) and a collision ($H_{2}$) entropiclower bound. Such an identification of smaller sets of MUBs satisfying tightEURs is crucial for cryptographic applications as well as constructing optimalentanglement witnesses for higher dimensional systems.

Unextendible mutually unbiased bases in prime-squared dimensions