For every n≥ 1, every unital stochastic map Phi:M_n→ M_n, and every normalized monotone Riemannian metric specified by an operator-monotone function kappa, the Riemannian contraction coefficient satisfies eta^Riem_kappa(Phi)=Lambda_2(Phi^daggerPhi), where Lambda_2(Phi^daggerPhi) is the Hilbert–Schmidt contraction on the traceless subspace. Equivalently, the contraction coefficient is independent of the choice of normalized monotone Riemannian metric.

Lesniewski–Ruskai Conjecture 4.9

For every $n\ge 1$ , every unital stochastic map $\Phi:M_n\to M_n$ , and every normalized monotone Riemannian metric specified by an operator-monotone function $\kappa$ , the Riemannian contraction coefficient satisfies $\eta^{\mathrm{Riem}}_{\kappa}(\Phi)=\Lambda_2(\Phi^{\dagger}\Phi)$ , where $\Lambda_2(\Phi^{\dagger}\Phi)$ is the Hilbert–Schmidt contraction on the traceless subspace. Equivalently, the contraction coefficient is independent of the choice of normalized monotone Riemannian metric.

Progress summary

Solved

An unrefereed preprint claims to disprove the conjecture with an explicit example in dimension three, which would settle the smallest failing dimension.

Lesniewski and Ruskai formulated Conjecture 4.9 in 1999, asserting that for unital stochastic maps the Riemannian contraction coefficient equals a quantity determined by the map and is therefore independent of the chosen monotone metric. The conjecture is now claimed to fail in dimension $3$ .

Known results

Hiai and Ruskai proved the conjectured identity for every unital qubit map, establishing that any counterexample among full matrix algebras must have dimension at least $3$ .

August 2026 counterexample

On August 16, 2026, Domingos S. P. Salazar’s unrefereed preprint gives an explicit doubly stochastic $3\times3$ matrix and associated unital entanglement-breaking channel, claiming for every normalized monotone Riemannian metric that $\eta_{\kappa}^{\mathrm{Riem}}(\Phi_K)>\Lambda_2(\Phi_K^\dagger\Phi_K)$ . This would disprove Conjecture 4.9 and show that dimension $3$ is minimal, but no independent verification or referee report was found.

Current status (as of August 2026): The qubit case is settled positively, while a dimension- $3$ counterexample is claimed in an unrefereed preprint and remains unverified.

Sources

arxiv.org

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Hilbert–Schmidt contraction formulation
For every unital stochastic map $\Phi:M_n\to M_n$ and every normalized monotone Riemannian metric, the metric contraction coefficient on traceless tangent vectors equals the Hilbert–Schmidt contraction coefficient on the traceless subspace, namely $\eta^{\mathrm{Riem}}_{\kappa}(\Phi)=\Lambda_2(\Phi^{\dagger}\Phi)$ .
source: Salazar, Domingos S. P., “A minimal qutrit counterexample to Conjecture 4.9 of Lesniewski and Ruskai”