Genuinely quantum SudoQ and its cardinality

• URL: http://arxiv.org/abs/2106.02967v2
• Date: Tue, 8 Jun 2021 16:14:29 GMT
• Title: Genuinely quantum SudoQ and its cardinality
• Authors: Jerzy Paczos, Marcin Wierzbi\'nski, Grzegorz Rajchel-Mieldzio\'c, Adam Burchardt, Karol \.Zyczkowski
• Abstract summary: We find the complete parameterization of the genuinely quantum solutions of $4 times 4$ SudoQ game. In particular, a solution with the maximal cardinality equal to 16 is presented.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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