Sharp joint scaling for centered bosonic Gaussian cloning

Characterize the sharp joint dependence on the number of modes and the target error for cloning centered pure bosonic Gaussian states, beyond the established linear-in-mode sample complexity at fixed error.

Background

The paper constructs an affine cloner for centered pure bosonic Gaussian states and proves that, for every fixed trace error, the required number of input copies is linear in the number of modes. The canonical representation-theoretic cloner instead has a quadratic dependence on the number of modes but better dependence on the precision parameter.

The authors explicitly state that the sharp combined dependence on the number of modes and the error parameter has not been determined. This is a refinement of the fixed-error result rather than a request to improve an already-resolved qualitative bound.

References

For centered bosonic states, the linear dependence on $n$ is optimal at each fixed error; the sharp joint dependence on $n$ and $\epsilon$ remains 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, Section 14

The exact minimum trace-distance error $\epsilon_{\mathrm{sq},1}\star(N,M)$ remains open; \cref{eq:ab-one-mode-trace-bracket} gives lower and upper bounds.

— Approximate cloning of structured pure states  (2610.06723 - Herasymenko et al., 5 Oct 2026) in Appendix, Section 14, subsection “One-mode optimality”