Conjecture on the coherent state rank of |1>^{⊗n}

Prove that the approximate coherent state rank of the n‑mode single‑photon Fock state |1⟩^{⊗n} equals 2^n, i.e., establish κ(|1⟩^{⊗n}) = 2^n.

Background

The paper proves a super‑polynomial lower bound κ(|1⟩{⊗n}) = n{Ω(log n)} by connecting coherent state decompositions to the border complexity of multi‑linear formulas for the permanent, while the best general upper bound obtained by tensoring single‑mode decompositions is 2n. Matching these would settle the precise scaling for this central multimode state and would have implications for classical simulation of Boson Sampling.

References

We conjecture that $\kappa(\ket1{\otimes n})=2n$.

— Lower Bounds on Coherent State Rank  (2604.00766 - Cottier et al., 1 Apr 2026) in Discussion and open questions