Entanglement required for logarithmic communication in the constructed function

Determine the minimum amount of shared entanglement required to achieve O(log n) communication for the total Boolean function constructed in Theorem 1, distinguishing whether the currently known lower bound of Ω(n^{1/3}) shared EPR pairs can be improved toward the Θ(n) pairs used by the protocol.

Background

The main theorem establishes an exponential separation for a family of total Boolean functions: entanglement-assisted one-way classical communication uses O(log n) bits, while unassisted one-way quantum communication requires Θ(n{1/3}) qubits. The protocol uses Θ(n) shared EPR pairs. A trade-off argument yields only an Ω(n{1/3}) lower bound on the number of shared EPR pairs needed for O(log n) communication, leaving the optimal entanglement requirement unresolved.

References

It remains open how much shared entanglement is necessary to achieve $O(\log n)$ communication for our function: our lower bound requires $\Omega(n{1/3})$ EPR pairs, whereas our protocol uses $\Theta(n)$.

— An exponential separation between entanglement-assisted and unassisted one-way quantum communication  (2610.02099 - Anselm et al., 1 Oct 2026) in Section 1, subsection “Our results”