Gaussian-Wigner distributions in quantum mechanics and optics

Simon, R.; Sudarshan, E. C. G.; Mukunda, N.

doi:10.1103/physreva.36.3868

Cited by 260 publications

(264 citation statements)

References 60 publications

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“…This property, which is implicit in the papers [39,40] by Simon et al, was proved by Narcowich in [24,25] (also see Narcowich and O'Connell [26]). It is easily checked using a characterization of the nonnegativity of + i 2 J .…”

Section: Derivation Of the Uncertainty Principlementioning

confidence: 80%

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

Gosson

2009

Found Phys

112

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We show that the strong form of Heisenberg's inequalities due to Robertson and Schrödinger can be formally derived using only classical considerations. This is achieved using a statistical tool known as the "minimum volume ellipsoid" together with the notion of symplectic capacity, which we view as a topological measure of uncertainty invariant under Hamiltonian dynamics. This invariant provides a right measurement tool to define what "quantum scale" is. We take the opportunity to discuss the principle of the symplectic camel, which is at the origin of the definition of symplectic capacities, and which provides an interesting link between classical and quantum physics.

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Section: Derivation Of the Uncertainty Principlementioning

confidence: 80%

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

Gosson

2009

Found Phys

112

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“…Analysing the entanglement properties of infinite dimensional systems is generally technically involved unless one restricts attention to so-called Gaussian states [1,9] , which are more easily described in phase space introducing the (Wigner-)characteristic function. Using the Weyl operator W ξ = e iξ T R for ξ ∈ R 2n we define the characteristic function as…”

Section: B Gaussian Statesmentioning

confidence: 99%

“…Gaussian states are exactly those states for which the characteristic function χ ρ is a Gaussian function in phase space [9] and are completely specified by their first and second moments, d and γ…”

Section: B Gaussian Statesmentioning

confidence: 99%

Manipulating quantum information by propagation

Perales¹,

Plenio²

2005

J. Opt. B: Quantum Semiclass. Opt.

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We study creation of bi-and multipartite continuous variable entanglement in structures of coupled quantum harmonic oscillators. By adjusting the interaction strengths between nearest neighbors we show how to maximize the entanglement production between the arms in a Y-shaped structure where an initial single mode squeezed state is created in the first oscillator of the input arm. We also consider the action of the same structure as an approximate quantum cloner. For a specific time in the system dynamics the last oscillators in the output arms can be considered as imperfect copies of the initial state. By increasing the number of arms in the structure, multipartite entanglement is obtained, as well as 1 → M cloning. Finally, we are considering configurations that implement the symmetric splitting of an initial entangled state. All calculations are carried out within the framework of the rotating wave approximation in quantum optics, and our predictions could be tested with current available experimental techniques.

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“…For thermal states of two similar but distinguishable oscillators each having mass m, angular frequency ω and coupling spring constant K with corresponding dimensionless coupling α = 2K/mω 2 , K = 0 and the values of L and M are given by [3,15]. We adopt (mω) −1/2 as our length unit.…”

Section: Example: Entanglement Of Gaussian Statesmentioning

confidence: 99%

“…This important set of states includes both thermal and 'squeezed' states of harmonic systems and plays a key role in several fields of theoretical and experimental physics; we use them as an example because their entanglement properties are better understood than those of other continuous-variable systems, while recognising that our approach is general. The corresponding configuration-space density matrix can be written [15] …”

Section: Example: Entanglement Of Gaussian Statesmentioning

confidence: 99%

Entanglement in general two-mode continuous-variable states: Local approach and mapping to a two-qubit system

Lin

Fisher

2007

Phys. Rev. A

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We present a new approach to the analysis of entanglement in smooth bipartite continuousvariable states. One or both parties perform projective filterings via preliminary measurements to determine whether the system is located in some region of space; we study the entanglement remaining after filtering. For small regions, a two-mode system can be approximated by a pair of qubits and its entanglement fully characterized, even for mixed states. Our approach may be extended to any smooth bipartite pure state or two-mode mixed state, leading to natural definitions of concurrence and negativity densities. For Gaussian states both these quantities are constant throughout configuration space.There has been growing interest in the quantification of entanglement in quantum systems. However, many systems of interest have continuous, rather than discrete, degrees of freedom [1]. The general characterization of entanglement in such continous-variable systems is a very difficult problem; most of the known results are for Gaussian states, where the logarithmic negativity [2] can be calculated for arbitrary bipartite divisions [3]. The entanglement of formation [4] is known exactly only for symmetric bipartite Gaussian states [5], and further measures of entanglement in multipartite Gaussian states are a topic of active current research [1,6]. For non-Gaussian states there is some progress in finding criteria for entanglement [7], but much less in quantifying it.In this letter we present a new approach to this problem, allowing entanglement to be quantified locally in arbitrary (including non-Gaussian) continuous-variable states. We concentrate on entanglement localized near particular regions in configuration space, which we analyse via a thought experiment in which the entangled state is first measured to localise it. This corresponds to a particular type of projective filtering, used to identify the distribution of entanglement in a state which has a preexisting bipartite structure. This contrasts with previous studies [8,9,10] of the entanglement of a finite region of space with the rest of the system. We show that in the limit where the sizes of the measured regions become very small the description of each mode in the system becomes isomorphic to a single qubit. In two-mode systems simple expressions result for entanglement monotones (concurrence and negativity), yielding natural definitions for corresponding densities in configuration space.The filtering process. Let Alice and Bob share a state of two distinguishable one-dimensional particles (or two modes). Alice can measure only the position of her particle or mode (coordinate q A ), Bob the position of his (coordinate q B ). They filter their state by determining whether or not the particles are found in particular regions of configuration space, and discard instances in which they are not. We refer to the resulting subensemble as the 'discarding ensemble'. For example, if Alice measures whether her coordinate is in a sub-region a, the corresponding projector iŝwhere...

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Gaussian-Wigner distributions in quantum mechanics and optics

Cited by 260 publications

References 60 publications

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

Manipulating quantum information by propagation

Entanglement in general two-mode continuous-variable states: Local approach and mapping to a two-qubit system

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