Quantum no-go results beyond quantum theory and enlarged operational settings

Determine whether analogous no-go results prohibiting total ordering hold in more general probabilistic theories, and establish the conditions under which total ordering can re-emerge when the operational setting is enlarged.

Background

The paper proves that every nontrivial quantum resource theory on a quantum system of dimension at least three contains incomparable states, even when the free operations are taken to be all free-state-preserving quantum channels. It also shows that this obstruction does not occur universally in classical systems, where explicit totally ordered resource theories exist in every finite dimension, and notes that total ordering can reappear in certain enlarged quantum settings such as asymptotic entanglement transformations and catalytic resource theories.

The authors leave unresolved whether the higher-dimensional quantum no-go phenomenon extends to more general probabilistic theories and what operational modifications are sufficient to restore total ordering. The question concerns settings beyond the deterministic, unassisted transformations on the full quantum state space treated in the paper.