Implication of global optimality for local optimality

Determine whether globally optimal multi-qubit reset dynamics under locally applied dissipative channels necessarily enforce optimality of the corresponding reduced single-qubit dynamics.

Background

The paper establishes a separation between global and local optimality: entanglement-assisted reset protocols may follow non-geodesic trajectories in the full multi-qubit state space while the reduced state of each qubit follows a geodesic path. This demonstrates that local optimality does not imply global optimality.

The converse implication remains unresolved. Specifically, the authors ask whether a globally optimal evolution would necessarily impose optimal reduced dynamics for the individual qubits, a question relevant to developing a comprehensive geometric theory of Mpemba-assisted reset.

References

Finally, it is worth explicitly noting an observation arising from our results: local optimality of a dynamics does not imply the same for the global evolution. This naturally raises the question as to whether the converse is true or not, i.e., does a globally optimal dynamics enforce local optimality?

— Geometric optimality of entanglement-induced fast qubit reset  (2608.26897 - Rinaldi et al., 27 Aug 2026) in Section 5, subsection “The role of entanglement,” and Section 6, “Conclusion and outlook”