Normal form decomposition for Gaussian-to-Gaussian superoperators

In this paper we explore the set of linear maps sending the set of quantum Gaussian states into itself. These maps are in general not positive, a feature which can be exploited as a test to check whether a given quantum state belongs to the convex hull of Gaussian states (if one of the considered ma...

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Bibliographic Details
Published in:arXiv.org 2015-05
Main Authors: De Palma, Giacomo, Mari, Andrea, Giovannetti, Vittorio, Holevo, Alexander S
Format: Article
Language:English
Subjects:
Characteristic functions
Convexity
Covariance matrix
Decomposition
Hulls
Normal distribution
Operators (mathematics)
Stretching
Online Access:Get full text
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Summary:In this paper we explore the set of linear maps sending the set of quantum Gaussian states into itself. These maps are in general not positive, a feature which can be exploited as a test to check whether a given quantum state belongs to the convex hull of Gaussian states (if one of the considered maps sends it into a non positive operator, the above state is certified not to belong to the set). Generalizing a result known to be valid under the assumption of complete positivity, we provide a characterization of these Gaussian-to-Gaussian (not necessarily positive) superoperators in terms of their action on the characteristic function of the inputs. For the special case of one-mode mappings we also show that any Gaussian-to-Gaussian superoperator can be expressed as a concatenation of a phase-space dilatation, followed by the action of a completely positive Gaussian channel, possibly composed with a transposition. While a similar decomposition is shown to fail in the multi-mode scenario, we prove that it still holds at least under the further hypothesis of homogeneous action on the covariance matrix.
ISSN:2331-8422
DOI:10.48550/arxiv.1502.01870