01758nas a2200157 4500008003900000022001400039245009500053210006900148260000800217490000700225520125900232100001801491700002901509700001401538856004801552 2017 d a2469-992600aScaling maps of s-ordered quasiprobabilities are either nonpositive or completely positive0 aScaling maps of sordered quasiprobabilities are either nonpositi caug0 v963 aContinuous-variable systems in quantum theory can be fully described through any one of the s-ordered family of quasiprobabilities Lambda(s) (a), s is an element of [-1,1]. We ask forwhat values of (s,a) is the scalingmap Lambda(s) (alpha) -{\textgreater} a(-2) Lambda(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 (s = 1,{\textbar}a{\textbar} {\textless}= 1) and (s = -1,{\textbar}a{\textbar} {\textgreater}= 1), and only in these two nontrivial 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 behavior of the scaling maps in the 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.1 aIvan, Solomon1 aSabapathy, Krishna Kumar1 aSimon, R. uhttps://grupsderecerca.uab.cat/giq/node/845