Mode-independent Gaussianity testing with single-copy measurements

Determine whether Gaussianity can be tested with mode-independent sample complexity using adaptive single-copy measurements, and establish whether coherent access to several copies is fundamentally necessary; characterize the tradeoff between sample complexity and quantum memory between single-copy protocols and unrestricted collective measurements.

Background

The paper establishes mode-independent sample complexity for testing pure fermionic and bosonic Gaussian states using genuinely multi-copy measurements: two copies for fermionic tests and two or three copies for the bosonic tests considered. These protocols therefore require coherent access to multiple copies, while the role of single-copy adaptive measurements remains unresolved.

The authors further identify a broader unresolved issue concerning the intermediate regime in which only bounded quantum memory is available. A solution would clarify whether the constant-sample guarantees depend essentially on coherent multi-copy access and would quantify the tradeoff between quantum memory and sample complexity.

References

This leaves several natural questions. First, what is possible with single-copy measurements? The tests studied here exploit genuinely multi-copy observables. Can Gaussianity still be tested with mode-independent sample complexity using adaptive single-copy measurements, or is coherent access to several copies fundamentally necessary? More generally, it would be interesting to characterize the tradeoff between sample complexity and quantum memory, interpolating between single-copy protocols and unrestricted collective measurements.

— Testing quantum Gaussianity with constant sample complexity  (2609.40270 - Rad et al., 30 Sep 2026) in Section 6, Conclusion and open questions (Section~\ref{sec:discussion})

Another interesting question is whether the present guarantees can be extended beyond pure inputs, and can the tolerant regime be sharpened? Our tolerant result applies only in the small-threshold regime $\varepsilon_A\le c\eta$. Determining the optimal relation between the two thresholds, and understanding which analogous guarantees remain possible for mixed inputs, would give a more complete picture of Gaussianity testing.

— Testing quantum Gaussianity with constant sample complexity  (2609.40270 - Rad et al., 30 Sep 2026) in Section 6, Conclusion and open questions (Section~\ref{sec:discussion})