Normal form decomposition for Gaussian-to-Gaussian superoperators

Abstract

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.

References in corpus (6)

• Quantifying the non-Gaussian character of a quantum state by quantum relative entropy
• Quantum Capacities of Bosonic Channels
• A measure of the non-Gaussian character of a quantum state
• One-mode Bosonic Gaussian channels: a full weak-degradability classification
• Directly estimating non-classicality
• Complete criterion for convex-Gaussian state detection

Cited by in corpus (2)

• Non-Markovianity Hierarchy of Gaussian Processes and Quantum Amplification
• On the optical fields propagation in realistic environments