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Genuinely Quantum Solutions of Orthogonal Sudoku Squares
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DOI：10.1002/qute.70293.png)
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Abstract
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Quantum Sudoku is the generalized version of the popular game Sudoku. A quantum Sudoku square is a special kind of quantum Latin square. A genuinely quantum solution of quantum Sudoku squares has been solved completely. In this paper, we introduce the notion of orthogonality of two quantum Sudoku squares. We establish three novel constructions of mutually orthogonal quantum Sudoku squares including Lorch construction, Pedersen-Vis construction and direct product construction. As a result, we determine the genuinely quantum solution of a pair of orthogonal Sudoku squares of size d 2 $d^2$ for any positive integer d ≥ 2 $d\ge 2$ . In general, we show that the resulting squares naturally define an absolutely maximally entangled state AME ( 4 , d 2 ) ${\rm AME}(4,d^2)$ . Furthermore, we also show that the mutually orthogonal quantum Sudoku squares give rise to mutually unbiased bases based on maximally entangled states in C q 2 ⊗ C q 2 $\mathbb {C}^{q^2}\otimes \mathbb {C}^{q^2}$ for any prime power q $q$ .
Keywords:
AME states
mutually unbiased bases
Quantum Latin squares
Quantum Sudoku squares
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