量子 Hedetniemi 猜想的一个反例
A counterexample to the quantum Hedetniemi conjecture
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- Institute for Quantum Information, RWTH Aachen University（亚琛工业大学量子信息研究所）
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中文总结 AI 辅助
本文构造显式有限图，否证量子 Hedetniemi 猜想，证明其量子色数乘积小于因子最小值，并给出更小反例及 Lean 4 形式化验证。
中文摘要 AI 辅助
Godsil、Roberson、Šámal 和 Severini 猜想：两个图的范畴积的量子色数等于各因子量子色数的最小值。我们否证了这一猜想：我们构造了显式的有限图 $G,H$，使得 \\[ \chi(G\times H) \leq 1538 < 1539 = \min(\chi_q(G),\chi_q(H)).\\] 这些图是通过使用一个补图的 Lovász theta 数（而不仅仅是分数色数）较大的基图，从 Zhu 对 Hedetniemi 猜想的反例中得到的。第一个因子的下界是 theta 界。对于第二个因子，我们将 Zhu 的论证改编到不交换的投影上：固定团颜色的步骤被算子之间的恒等式所取代。两个下界对于任意非零单位 $C^*$-代数中的投影着色都成立。因此，该猜想对于量子色数的空间、近似、交换算子以及 $C^*$-代数变体也均不成立。我们还给出了由精确整数数据验证的更小的反例。图构造、证书以及投影表述中的反例陈述已在 Lean 4 中形式化。
英文摘要
Godsil, Roberson, Šámal and Severini conjectured that the quantum chromatic number of the categorical product of two graphs equals the minimum of the quantum chromatic numbers of the factors. We disprove this conjecture: we construct explicit finite graphs $G,H$ with \[ χ(G\times H) \leq 1538 < 1539 = \min(χ_q(G),χ_q(H)).\]The graphs are obtained from Zhu's counterexample to Hedetniemi's conjecture by using a base graph for which the Lovász theta number of the complement, and not only the fractional chromatic number, is large. The lower bound for the first factor is the theta bound. For the second factor we adapt Zhu's argument to projections that do not commute: the step that fixes the colors of a clique is replaced by identities between operators. Both lower bounds hold for colorings by projections in an arbitrary nonzero unital $C^*$-algebra. Hence the conjecture also fails for the spatial, approximate, commuting-operator and $C^*$-algebraic variants of the quantum chromatic number. We also give smaller counterexamples certified by exact integer data. The graph constructions, the certificates and the counterexample statements in the projective formulation are formalized in Lean~4.
