bibkey: paczosetal2021sudoq authors: J. Paczos, M. Wierzbiński, G. Rajchel-Mieldzioć, A. Burchardt, K. Życzkowski year: 2021 title: “Genuinely quantum SudoQ and its cardinality” doi: 10.1103/PhysRevA.104.042423 url: https://arxiv.org/abs/2106.02967v2 claim: “Section III.A conjectures that no 4×4 SudoQ with three clues is uniquely solvable; Theorem 13 gives uniquely solvable grids with four clues.” strata_touched:
- D5/S3/Quantum/LatinSquares/SudoQClueGeometry
- D5/S3/Quantum/LatinSquares/SudoQMinimumClues license: citation-only triage: anchor
Verified locator
DOI: 10.1103/PhysRevA.104.042423
Source: https://arxiv.org/abs/2106.02967v2
Published as “Genuinely quantum solutions of the game Sudoku and their cardinality”, Physical Review A 104 (2021), 042423.
Source statements
Page numbers refer to arXiv:2106.02967v2.
Definition 2, p. 2:
A quantum Latin square (QLS) is an array of vectors from -dimensional Hilbert space , arranged in such a way that each row and column forms an orthonormal basis of .
Definition 7, p. 3:
A quantum Sudoku (SudoQ) is an QLS with additional constraints that each of the disjoint blocks of size forms an orthonormal basis.
Definition 8, p. 3:
A quantum grid is a SudoQ grid whose some entries can be blank.
Definition 9, p. 3:
A quantum grid is called solvable if it has at least one solution.
Definition 10, p. 3:
A quantum grid is called uniquely solvable if it has a unique solution.
Definition 5, p. 2:
The cardinality c of a QLS is the number of its vectors distinct up to a global phase.
Section III.A, p. 4:
Conjecture. Similarly to the classical Sudoku there are no 4×4 SudoQ with 3 clues that are uniquely solvable.
Theorem 13, p. 4, gives a uniquely solvable four-clue grid for every 4×4 quantum square of cardinality 16; its displayed clues are e1, f3, v2 and u4 at the four positions (1,1), (2,3), (3,2) and (4,4), using the paper’s one-based indices.
G. Rajchel-Mieldzioć, Quantum mappings and designs, https://arxiv.org/abs/2204.13008, Chapter 7, restates the conjecture:
Uniquely solvable grids of 4 × 4 SudoQ admit at least 4 clues.
Encoding
The specialization is N = 2. Vectors are EuclideanSpace ℂ (Fin 4). Rows and columns use zero-based Fin 4 indices. Blocks have indices b : Fin 2 × Fin 2 and offsets k : Fin 2 × Fin 2, with cell (2b.1 + k.1, 2b.2 + k.2). Four orthonormal vectors span the four-dimensional space. A finite set C marks the nonblank entries, with literal prescribed vectors v on C; values away from C impose no constraints. Solutions are compared entry by entry up to a unit complex phase, as rays. Existence of a solution is part of unique solvability. The statement with at most three clues implies the paper’s statement with exactly three.
Mathematical consequence and boundary
The lower bound, together with the source’s Theorem 13, gives an exact minimum of four clues for uniquely solvable 4×4 SudoQ. The proof uses clue-fixing reflection and basis swaps, followed by ray normalization and a four-cell trade avoiding the clues. It includes unsolvable grids and arbitrary complex clue vectors. The source’s cardinality classification and four-clue construction are unaffected. The minimum number of clues for 9×9 SudoQ is open.
The finite experiment is
check.py,
sha256 e37ab13c1f3122f7063ceed89cb700c075970ebf88d9bdb23624fdb5476a83d8.
Its reported command is python3 check.py, exit 0, final reading bad= 0.
This computed check is separate from the Lean proof.
Literature boundary
No earlier settlement was located in the inspected arXiv works and citation abstracts. The PRA full text and the publisher full text of DOI 10.1002/qute.70293 were not verified. Coverage of unindexed later publications is ASSUMED-UNVERIFIED; absence of an earlier settlement is not a kernel claim.