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

Scaling maps of s-ordered quasiprobabilities are either nonpositive or completely positive Continuous-variable systems in quantum theory can be fully described through any one of the s-ordered family of quasiprobabilities Λs(α), s∈[−1,1]. We ask for what values of (s,a) is the scaling map Λs(α)→a−2Λs(a−1α) 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,|a|≤1) and (s=−1,|a|≥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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review A American Physical Society (APS)

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

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

Abstract

Continuous-variable systems in quantum theory can be fully described through any one of the s-ordered family of quasiprobabilities Λs(α), s∈[−1,1]. We ask for what values of (s,a) is the scaling map Λs(α)→a−2Λs(a−1α) 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,|a|≤1) and (s=−1,|a|≥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.
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Publisher
The American Physical Society
Copyright
Copyright © ©2017 American Physical Society
ISSN
1050-2947
eISSN
1094-1622
D.O.I.
10.1103/PhysRevA.96.022114
Publisher site
See Article on Publisher Site

Abstract

Continuous-variable systems in quantum theory can be fully described through any one of the s-ordered family of quasiprobabilities Λs(α), s∈[−1,1]. We ask for what values of (s,a) is the scaling map Λs(α)→a−2Λs(a−1α) 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,|a|≤1) and (s=−1,|a|≥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.

Journal

Physical Review AAmerican Physical Society (APS)

Published: Aug 9, 2017

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