Optimal trace-distance cloning of phase states

Construct or characterize cloners with optimal trace-distance error for the qudit phase-state family, and determine its exact high-precision trace-distance cloning complexity.

Background

The paper determines a channel that is optimal for global fidelity when cloning qudit phase states and derives fidelity-based sample-complexity bounds. Trace distance, however, is the primary error criterion used elsewhere in the paper, and fidelity optimality does not generally imply trace-distance optimality.

The authors explicitly leave both the exact high-precision trace-distance complexity and the existence or characterization of trace-distance-optimal cloners unresolved.

References

For phase states, the channels presented in this work have optimal fidelity and provide good bounds for complexity in terms of trace distance. However, this complexity in the high-precision regime is not exactly known, and cloners with optimal trace distance remain an open question.

— Approximate cloning of structured pure states  (2610.06723 - Herasymenko et al., 5 Oct 2026) in Section 7, subsection “Outlook”