  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, which are 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
parameterization of the genuinely quantum solutions of $4 \times 4$ 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 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.

Quantum Physics