Optimality of Adaptive Fidelity Testing Algorithms

Determine whether the presented adaptive sample complexities for quantum-state equivalence testing and independence testing in fidelity are optimal, and whether adaptive equivalence testing in fidelity is fundamentally harder than certification.

Background

The paper develops adaptive single-copy algorithms for equivalence testing and independence testing in fidelity, with sample complexities that improve over generic trace-distance testing in some parameter regimes. The authors do not establish matching lower bounds for these adaptive algorithms.

The unresolved issue includes both the optimality of the stated sample complexities and the possible intrinsic separation between equivalence testing, where both states are unknown, and certification, where one state is known.

References

While it remains an open question whether our algorithms for equivalence testing and independence testing are optimal, we conjecture that optimal approaches will need to take advantage of the different problem settings in a similar manner, which we in particular expect to be reflected in different sample complexities.

— On the Power of Adaptivity in Testing Quantum States in Fidelity  (2609.08733 - Seyfried et al., 8 Sep 2026) in Introduction, subsection Discussion

We therefore state the following open questions: \begin{itemize} \item If adaptivity is not allowed, is there an asymptotic advantage in equivalence testing or independence testing in fidelity over trace distance? \item Are the presented sample complexities for equivalence testing in fidelity and independence testing optimal, and if not, is (adaptive) equivalence testing generally harder than certification? \item Are instance-optimal bounds for certification in fidelity analogous to the existing results for trace distance? \end{itemize}

— On the Power of Adaptivity in Testing Quantum States in Fidelity  (2609.08733 - Seyfried et al., 8 Sep 2026) in Introduction, subsection Discussion