Tight bounded-error lower bound for multiparty Index Coordination

Determine whether, for each fixed k, the bounded-error quantum communication lower bound for the multiparty Index Coordination relation can be improved from Ω(n^{(k−1)/(k+1)}) to the unambiguous-regime bound Ω(n^{1−1/k}), or whether an unentangled quantum protocol with O(n^{(k−1)/(k+1)}) communication exists.

Background

For fixed k, the paper establishes different lower bounds in the two error regimes: Ω(n{1−1/k}) for unambiguous protocols and Ω(n{(k−1)/(k+1)}) for bounded-error protocols. The discrepancy results from using an L1 constraint on zero-error local identification probabilities in the unambiguous case and an L2 constraint on trace-distance biases in the bounded-error case.

The authors state that it is unresolved whether this gap reflects an intrinsic difference between the regimes or merely a limitation of the proof. They formulate two alternatives: strengthening the bounded-error lower bound to match the unambiguous bound, or constructing an unentangled quantum protocol that achieves communication at the currently established bounded-error exponent.

References

Whether this difference reflects the complexity of the problem or a limitation of the proof remains open.

For fixed $k$, can the bounded-error quantum lower bound for $\operatorname{IC}_{k,n}$ be improved to match the $\Omega(n{1-1/k})$ unambiguous bound, or does there exist an unentangled quantum protocol that exploits the permitted error margin to achieve $O\left(n{\frac{k-1}{k+1}\right)$ communication?

— On the Limits of Quantum Multiparty Simultaneous Communication  (2609.10289 - Montealegre et al., 9 Sep 2026) in Section 5, paragraph “Unifying the Lower Bounds Across Error Regimes”