Normal form decomposition for Gaussian-to-Gaussian superoperators

Palma, Giacomo De; Mari, Andrea; Giovannetti, Vittorio; Holevo, Alexander Semenovich

doi:10.1063/1.4921265

Cited by 27 publications

(20 citation statements)

References 34 publications

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“…[101] and [102]. Non-GKSL generators lead to positive but noncompletely positive maps which are also actively discussed [103,104,105,107,108]. The master equations with time-non-local generators and their relations to the time-local equations are also intensively studied now [109,110,111,112].…”

Section: Resultsmentioning

confidence: 99%

Irreversible quantum evolution with quadratic generator: Review

Teretenkov

2019

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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We review results on GKSL-type equations with multi-modal generators which are quadratic in bosonic or fermionic creation and annihilation operators. General forms of such equations are presented. The Gaussian solutions are obtained in terms of equations for the first and the second moments. Different approaches for their solutions are discussed.was developed in the works by I. A. Malkin, V. I. Man'ko and V. V. Dodonov [3,4,5]. Some modern studies on unitary dynamics with quadratic Hamiltonians could be found in works by A. M. Chebotraev and T. V. Tlyachov [7,8,9]. One of the main areas of application of quadratic multi-modal Hamiltonians is quantum optics, especially in case of parametric approximation (see Ref.[10], Ch. II, Ref. [11], Ch. 16, or Ref. [12], Sec. 7.1.2). Coherent states for multi-modal parametric systems were considered in Ref. [13]. Modern applications could be found in works by A. S. Chirkin et al [14,15,16,17,18,19]. Also quadratic Hamiltonians arise in opto-mechanical problems [20,21]. Another source of such Hamiltonians is the approximate quantization method by N. N. Bogolyubov [22]. (See also Ref. 80 for contemporary discussion.) Discussion of the unitary dynamics with non-stationary (time-dependent) Hamiltonians could be found in Refs.[24]- [26]. Physical applications of such Hamiltonians were discussed in Ref. [27]. Quantum evolution with a quadratic generator is also closely related to the classical one, which was studied in the general case by J. Williamson [28,29,30]. The quasi-classical approximation for squeezed states was discussed in Ref. [31]. More detailed bibliography for bosonic unitary evolution could be found in Refs. [5] and [32]. The fermionic case without dissipation was considered in Ref.[2] by F. A. Berezin. The unified approach both for the bosonic reversible dynamics and the fermionic one was considered by V. I. Man'ko and V. V. Dodonov.[33] We consider reversible (non-unitary) evolution with quadratic generators. From the historical point of view one should mention Ref.[34] by L. D. Landau, which is famous for the first appearance of the concept of the density matrix. What is interesting for this review is that equations for irreversible quantum evolution of the density matrix were also introduced there. As it was mentioned on p. 183 in Ref.[5] and in Sec. 23.1 of Ref. [35] in the harmonic oscillator case one has (in modern notation)ρ

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Section: Resultsmentioning

confidence: 99%

Irreversible quantum evolution with quadratic generator: Review

Teretenkov

2019

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…These maps Φ have been studied in details in Ref. [76], where it is proven that they are characterised by a finite number of parameters. Therefore, again a parameter-counting argument proves the claim.…”

Section: Absence Of Maximally Resourceful Statesmentioning

confidence: 99%

Resource theory of quantum non-Gaussianity and Wigner negativity

et al. 2018

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We develop a resource theory for continuous-variable systems grounded on operations routinely available within current quantum technologies. In particular, the set of free operations is convex and includes quadratic transformations and conditional coarse-grained measurements. The present theory lends itself to quantify both quantum non-Gaussianity and Wigner negativity as resources, depending on the choice of the free-state seti.e., the convex hull of Gaussian states or the states with positive Wigner function, respectively. After showing that the theory admits no maximally resourceful state, we define a computable resource monotone -the Wigner logarithmic negativity. We use the latter to assess the resource content of experimentally relevant states -e.g., photon-added, photon-subtracted, cubic-phase, and cat states -and to find optimal working points of some resource concentration protocols. We envisage applications of this framework to sub-universal and universal quantum information processing over continuous variables. * francesco.albarelli@unimi.it † marco.genoni@fisica.unimi.it ‡ matteo.paris@fisica.unimi.it § a.ferraro@qub.ac.uk arXiv:1804.05763v2 [quant-ph] 4 Dec 2018 1. M(ρ) = 0 ∀ρ ∈ G (resp. W + ).

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“…(6) φ ∈ X G , then the original linear CP map T is also Gaussian. And if φ is linear, the requirement in Definition 2 is equivalent to the weaker condition: ∀ρ G ∈ G[n φ ], we have φ (ρ G ) ∈ G [63,64]. Since on Gaussian inputs, Gaussian measurements can also be transformed to TP operations by post-processing [7], we are particularly interested in the set of Gaussian channels X L G ⊂ X G .…”

Section: Gaussian Operationsmentioning

confidence: 99%

Resource theory of non-Gaussian operations

2018

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Non-Gaussian states and operations are crucial for various continuous-variable quantum information processing tasks. To quantitatively understand non-Gaussianity beyond states, we establish a resource theory for non-Gaussian operations. In our framework, we consider Gaussian operations as free operations, and non-Gaussian operations as resources. We define entanglement-assisted non-Gaussianity generating power and show that it is a monotone that is non-increasing under the set of free super-operations, i.e., concatenation and tensoring with Gaussian channels. For conditional unitary maps, this monotone can be analytically calculated. As examples, we show that the non-Gaussianity of ideal photon-number subtraction and photon-number addition equal the non-Gaussianity of the single-photon Fock state. Based on our non-Gaussianity monotone, we divide non-Gaussian operations into two classes: (1) the finite non-Gaussianity class, e.g., photon-number subtraction, photon-number addition and all Gaussian-dilatable non-Gaussian channels; and (2) the diverging non-Gaussianity class, e.g., the binary phase-shift channel and the Kerr nonlinearity. This classification also implies that not all non-Gaussian channels are exactly Gaussian-dilatable. Our resource theory enables a quantitative characterization and a first classification of non-Gaussian operations, paving the way towards the full understanding of non-Gaussianity.

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Normal form decomposition for Gaussian-to-Gaussian superoperators

Cited by 27 publications

References 34 publications

Irreversible quantum evolution with quadratic generator: Review

Irreversible quantum evolution with quadratic generator: Review

Resource theory of quantum non-Gaussianity and Wigner negativity

Resource theory of non-Gaussian operations

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