# Scaling maps of $s$-ordered quasiprobabilities are either nonpositive or completely positive

## TL;DR

This paper investigates when scaling transformations of s-ordered quasiprobability distributions in continuous-variable quantum systems are positive or completely positive, revealing that such maps are generally nonpositive except in specific cases linked to noiseless channels.

## Contribution

The study characterizes the positivity of scaling maps for s-ordered quasiprobabilities, identifying precise conditions under which they are positive or completely positive, and provides a phase diagram in parameter space.

## Key findings

- Scaling maps are mostly nonpositive, indicating limited use as entanglement witnesses.
- Only in specific cases related to noiseless channels are the maps completely positive.
- A phase diagram illustrates the positivity behavior across parameters.

## Abstract

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.