Linear-versus-quadratic complexity for all pure bosonic Gaussian states

Determine whether the quadratic upper bound in the number of modes for cloning the full family of pure bosonic Gaussian states can be improved to a linear bound.

Background

For the full family of pure bosonic Gaussian states, in which both displacement and squeezing are unknown, the paper proves lower and upper sample-complexity bounds that are respectively linear and quadratic in the number of modes. The canonical cloner supplies the quadratic upper bound.

The unresolved issue is whether the quadratic dependence is intrinsic or can be reduced to linear dependence, which would close the principal complexity gap for this family.

References

For the full Gaussian family, whether the quadratic upper bound can be improved to linear in $n$ is also open.

— Approximate cloning of structured pure states  (2610.06723 - Herasymenko et al., 5 Oct 2026) in Section 2, subsection “Gaussian states and resolving orbits”; Section 7, Theorem “Cloning all pure Gaussian states”; Appendix, Section 14