Characterization of encodings with non-Hermitian local factors
Characterize the quantum states that can be encoded by pairwise-commuting product unitaries whose local factors are not binary Hermitian observables, thereby extending the commuting-reflection characterization.
References
Commuting encoders whose local factors are not binary observables are not covered by the characterization, and identifying the states they can encode is one of the open problems we highlight in \cref{sec:discussion}.
This immediately raises what we consider the central open question. Which other encoding classes define natural state classes, and what do these sets look like? Concrete intermediate classes could be: coordinated tables of Pauli operators that need not commute, as in the $W$-basis of Miyake and Briegel ; commuting families whose local factors are not Hermitian, the smallest step beyond our theorem; and tables of local Clifford unitaries, as in our $M_4$ construction. For none of these is a characterization known, although $M_4$ now proves that the last class is strictly larger than commuting reflections.
This suggests a stronger form of the positive numerical picture of Ref.: \emph{Conjecture.---Every three-qubit pure state admits a commuting-reflection local encoding.}