A minimal qutrit counterexample to Conjecture 4.9 of Lesniewski and Ruskai

Abstract

Lesniewski and Ruskai conjectured that the contraction coefficient of every monotone Riemannian metric under a unital stochastic map equals the Hilbert--Schmidt contraction on the traceless subspace. We disprove the conjecture with an explicit entanglement-breaking qutrit channel induced by a doubly stochastic $3\times3$ matrix. A faithful diagonal state and a commuting traceless tangent give, simultaneously for every normalized monotone metric, the exact lower bound $\eta^{\mathrm{Riem}}_{\kappa}(\Phi_K)\ge 8896/20007>(62+2\sqrt{61})/225=\Lambda_2(\Phi_K^{\dagger}\Phi_K)$. The counterexample is entirely classical on a maximal abelian subalgebra. A theorem of Hiai and Ruskai establishes the conjectured identity for all unital qubit maps, so dimension three is minimal among full matrix algebras.