Genuinely quantum SudoQ and its cardinality

TL;DR

This paper introduces the concept of cardinality in quantum Sudoku (SudoQ), characterizes genuinely quantum solutions for 4x4 grids, and proves the existence of solutions with maximal cardinality, connecting to mutually unbiased bases.

Contribution

It provides a complete parameterization of genuinely quantum solutions for 4x4 SudoQ and proves the existence of solutions with maximal cardinality for any size, linking to mutually unbiased bases.

Findings

Admissible cardinalities for 4x4 SudoQ are 4, 6, 8, and 16.

Constructed a solution with maximal cardinality 16.

Proved the existence of solutions with cardinality N^4 for any N, related to mutually unbiased bases.

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 $N^2$ 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.