# 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:

- NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa (Italy)
- (Italy)
- 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}

}