  Continuous-variable systems in quantum theory can be fully described through
any one of the ${\rm s}$-ordered family of quasiprobabilities $\Lambda_{\rm
s}(\alpha)$, ${\rm s} \in [-1,1]$. We ask for what values of $({\rm s}, a)$ is
the scaling map $\Lambda_{\rm s}(\alpha) \rightarrow a^{-2} \Lambda_{\rm
s}(a^{-1}\alpha)$ a positive map? Our analysis based on a duality we establish
settles this issue (i) the scaling map generically fails to be positive,
showing that there is no useful entanglement witness of the scaling type beyond
the transpose map, and (ii) in the two particular cases $({\rm s}=1, |a| \leq
1)$ and $({\rm s}=-1, |a| \geq 1)$, and only in these two non-trivial cases,
the map is not only positive but also completely positive as seen through the
noiseless attenuator and amplifier channels. We also present a `phase diagram'
for the behaviour of the scaling maps in the ${\rm s}-a$ parameter space with
regard to its positivity, obtained from the viewpoint of symmetric-ordered
characteristic functions. This also sheds light on similar diagrams for the
practically relevant attenuation and amplification maps with respect to the
noise parameter, especially in the range below the complete-positivity (or
quantum-limited) threshold.
