Structural criteria for exponential decomposition complexity

Identify structural properties of a family of free states that ensure exponential decomposition complexity for tensor powers of every fixed pure state outside that family.

Background

The results show that non-Gaussianity forces exponential Gaussian decomposition complexity, but the paper does not characterize which structural features of a general family of free states are sufficient to produce an analogous result. The authors leave open the broader task of identifying such properties beyond the specific Gaussian setting.

References

More generally, it remains to identify additional structural properties of a family of free states that ensure exponential decomposition complexity for tensor powers of every fixed pure state outside the family.

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