Genuinely quantum SudoQ and its cardinality

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.

References in corpus (11)

• Security of quantum key distribution using d-level systems
• A new proof for the existence of mutually unbiased bases
• Operational Significance of Discord Consumption: Theory and Experiment
• Three ways to look at mutually unbiased bases
• Entanglement and quantum combinatorial designs
• Constructing Mutually Unbiased Bases from Quantum Latin Squares
• Entanglement Patterns in Mutually Unbiased Basis Sets for N Prime-state Particles
• On bipartite unitary matrices generating subalgebra-preserving quantum operations
• Detecting Entanglement can be More Effective with Inequivalent Mutually Unbiased Bases
• Orthogonality for Quantum Latin Isometry Squares
• Quantum magic squares: dilations and their limitations

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• Mutually unbiased maximally entangled bases from difference matrices
• Quantum permutation puzzles with indistinguishable particles
• Absolutely maximally entangled pure states of multipartite quantum systems