Abstract
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 N² that have cardinality greater than N², and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parameterization of the genuinely quantum solutions of 4 × 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 N² × N² SudoQ solution of cardinality N⁴, which for a prime N is related to a set of N mutually unbiased bases of size N². Such a construction of N⁴ different vectors of size N yields a set of N³ orthogonal measurements.