Universal compatibility of ordering-family criteria

Determine whether some member of the split-ordering family \(\Phi_s(E,\rho)=\tfrac12(\rho^sE\rho^{1-s}+\rho^{1-s}E\rho^s)\) can satisfy every reasonable condition considered for a quantum smoothed state simultaneously.

Background

The paper studies a family of candidate smoothed-state operators obtained by symmetrizing the split orderings of a filtered state $\rho$ and a post-selection effect $E$. The symmetric member $\Phi_{1/2}=\rho^{1/2}E\rho^{1/2}$ is positive for all states and effects and is also singled out by multiplicativity, while other criteria and physical considerations do not establish a unique universally satisfactory smoothed state.

The authors explicitly leave unresolved whether any single member of the ordering family can meet all reasonable conditions at once, including the competing criteria discussed in the paper and its Supplemental Material.