skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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.

Authors:
 [1];  [2]; ;  [1];  [3]
  1. NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa (Italy)
  2. (Italy)
  3. Steklov Mathematical Institute, 119991 Moscow, Russia and National Research University Higher School of Economics (HSE), 101000 Moscow (Russian Federation)
Publication Date:
OSTI Identifier:
22403144
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DECOMPOSITION; MAPPING; MAPS; PHASE SPACE; QUANTUM STATES; SUPEROPERATORS

Citation Formats

De Palma, Giacomo, INFN, Pisa, Mari, Andrea, Giovannetti, Vittorio, and Holevo, Alexander S. Normal form decomposition for Gaussian-to-Gaussian superoperators. United States: N. p., 2015. Web. doi:10.1063/1.4921265.
De Palma, Giacomo, INFN, Pisa, Mari, Andrea, Giovannetti, Vittorio, & Holevo, Alexander S. Normal form decomposition for Gaussian-to-Gaussian superoperators. United States. doi:10.1063/1.4921265.
De Palma, Giacomo, INFN, Pisa, Mari, Andrea, Giovannetti, Vittorio, and Holevo, Alexander S. Fri . "Normal form decomposition for Gaussian-to-Gaussian superoperators". United States. doi:10.1063/1.4921265.
@article{osti_22403144,
title = {Normal form decomposition for Gaussian-to-Gaussian superoperators},
author = {De Palma, Giacomo and INFN, Pisa and Mari, Andrea and Giovannetti, Vittorio and Holevo, Alexander S.},
abstractNote = {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.},
doi = {10.1063/1.4921265},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 5,
volume = 56,
place = {United States},
year = {2015},
month = {5}
}