Multimode Uncertainty Relations and Separability of Continuous Variable States

Serafini, Alessio

doi:10.1103/physrevlett.96.110402

Cited by 110 publications

(136 citation statements)

References 34 publications

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“…The N symplectic eigenvalues are thus determined by N invariants of the characteristic polynomial of the matrix |iΩσ| [38]. Knowing the symplectic spectrum of a given covariance matrix is very powerful.…”

Section: Symplectic Geometry and The Williamson Theoremmentioning

confidence: 99%

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Continuous Variable Quantum Information: Gaussian States and Beyond

Adesso

Ragy

Lee

2014

Open Syst. Inf. Dyn.

683

889

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The study of Gaussian states has arisen to a privileged position in continuous variable quantum information in recent years. This is due to vehemently pursued experimental realisations and a magnificently elegant mathematical framework. In this article, we provide a brief, and hopefully didactic, exposition of Gaussian state quantum information and its contemporary uses, including sometimes omitted crucial details. After introducing the subject material and outlining the essential toolbox of continuous variable systems, we define the basic notions needed to understand Gaussian states and Gaussian operations. In particular, emphasis is placed on the mathematical structure combining notions of algebra and symplectic geometry fundamental to a complete understanding of Gaussian informatics. Furthermore, we discuss the quantification of different forms of correlations (including entanglement and quantum discord) for Gaussian states, paying special attention to recently developed measures. The manuscript is concluded by succinctly expressing the main Gaussian state limitations and outlining a selection of possible future lines for quantum information processing with continuous variable systems.

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Section: Symplectic Geometry and The Williamson Theoremmentioning

confidence: 99%

“…We note that such a constraint implies σ > 0. Inequality (29) is the expression of the uncertainty principle on the canonical operators in its strong, Robertson-Schrödinger form [36,37,38]. Gaussian states ρ can, of course, be pure or mixed.…”

Section: Andmentioning

confidence: 99%

Continuous Variable Quantum Information: Gaussian States and Beyond

Adesso

Ragy

Lee

2014

Open Syst. Inf. Dyn.

683

889

View full text Add to dashboard Cite

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“…In addition we compare our results to Dodonov's and Sera…ni's "quantum universal invariants" [3,14,15], which comforts us in our belief that the "Angel of Geometry"should always be preferred to the "Demon of Algebra" in conceptual questions.…”

Section: Introductionmentioning

confidence: 77%

“…In [14,15] Sera…ni studies the "universal symplectic invariants" introduced by Dodonov [3]. Working in units in which~= 2, he denotes by n j the principal minor of order 2j of , i.e., the sum of the determinants of all the principal submatrices of order 2j of (by convention n 0 = 1).…”

Section: Discussionmentioning

confidence: 99%

The Symplectic Camel and Quantum Universal Invariants: The Angel of Geometry versus the Demon of Algebra

Gosson¹

2014

Quant. Matt

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A positive de…nite symmetric matrix quali…es as a quantum mechanical covariance matrix if and only if + 1 2 i~ 0 where is the standard symplectic matrix. This well-known condition is a strong version of the uncertainty principle, which can be reinterpreted in terms of the topological notion of symplectic capacity, closely related to Gromov's non-squeezing theorem. We show that a recent re…nement of the latter leads to a new class of geometric invariants. These are the volumes of the orthogonal projections of the covariance ellipsoid on symplectic subspaces of the phase space. We compare these geometric invariants to the algebraic "universal quantum invariants"of Dodonov and Sera…ni.

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“…Note that the previous condition is necessary for the CM of any (generally non Gaussian) state, as it generalizes to many modes the Robertson-Schrödinger uncertainty relation [10]. A major role in the theoretical and experimental manipulation of Gaussian states is played by unitary operations which preserve the Gaussian character of the states on which they act.…”

Section: Continuous Variable Systems and Gaussian Statesmentioning

confidence: 99%

Optical implementation and entanglement distribution in Gaussian valence bond states

Adesso

Ericsson

2007

Opt. Spectrosc.

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Abstract. We study Gaussian valence bond states of continuous variable systems, obtained as the outputs of projection operations from an ancillary space of M infinitely entangled bonds connecting neighboring sites, applied at each of N sites of an harmonic chain. The entanglement distribution in Gaussian valence bond states can be controlled by varying the input amount of entanglement engineered in a (2M + 1)-mode Gaussian state known as the building block, which is isomorphic to the projector applied at a given site. We show how this mechanism can be interpreted in terms of multiple entanglement swapping from the chain of ancillary bonds, through the building blocks. We provide optical schemes to produce bisymmetric three-mode Gaussian building blocks (which correspond to a single bond, M = 1), and study the entanglement structure in the output Gaussian valence bond states. The usefulness of such states for quantum communication protocols with continuous variables, like telecloning and teleportation networks, is finally discussed. Optical implementation and entanglement distribution in Gaussian valence bond states2

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Multimode Uncertainty Relations and Separability of Continuous Variable States

Cited by 110 publications

References 34 publications

Continuous Variable Quantum Information: Gaussian States and Beyond

Continuous Variable Quantum Information: Gaussian States and Beyond

The Symplectic Camel and Quantum Universal Invariants: The Angel of Geometry versus the Demon of Algebra

Optical implementation and entanglement distribution in Gaussian valence bond states

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