Optimal approximate-border Gaussian-rank growth rates

Determine the optimal exponential growth rates of the approximate border Gaussian ranks of tensor powers of the four-mode fermionic state |M⟩ = (|0000⟩ + |1111⟩)/√2 and the bosonic single-photon state |1⟩, including whether these rates are independent of the approximation error.

Background

The paper establishes exponential lower bounds for the approximate border Gaussian rank of tensor powers of the canonical four-mode fermionic GHZ state and the bosonic single-photon state. However, the derived lower-bound bases are not shown to be optimal, and matching upper bounds are not established. The authors explicitly identify determining the exact exponential rates and their dependence on the allowed approximation error as an unresolved question.

References

For |M⟩{\otimes k} and |1⟩{\otimes k}, determining the optimal exponential growth rates of approximate border Gaussian rank requires matching upper and lower bounds and understanding whether these rates are independent of approximation error.

— Robust exponential lower bounds for fermionic and bosonic Gaussian ranks  (2610.02172 - Wei et al., 1 Oct 2026) in Section Discussion and outlook