We expand the quantum variant of the popular game Sudoku by introducing the notion of cardinality of a quantum Sudoku (SudoQ), equal to the number of distinct vectors appearing in the pattern. Our considerations are focused on the genuinely quantum solutions—the solutions of size $N^2$ that have cardinality greater than $N^2$, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parametrization of the genuinely quantum solutions of a $4\times 4$ SudoQ game and establish that in this case the admissible cardinalities are 4, 6, 8, and 16. In particular, a solution with the maximal cardinality equal to 16 is presented. Furthermore, the parametrization enabled us to prove a recent conjecture of Nechita and Pillet [I. Nechita and J. Pillet, Quantum Inf. Comput. 21, 781 (2021)] for this special dimension. In general, we proved that for any $N$ it is possible to find an $N^2\times N^2$ SudoQ solution of cardinality $N^4$, which for a prime $N$ is related to a set of $N$ mutually unbiased bases of size $N^2$. Such a construction of $N^4$ different vectors of size $N$ yields a set of $N^3$ orthogonal measurements.

• Received 13 July 2021
• Accepted 20 September 2021

DOI:https://doi.org/10.1103/PhysRevA.104.042423