On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems

Williamson, John

doi:10.2307/2371062

Cited by 701 publications

(505 citation statements)

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“…average number of photons, and U S denotes the unitary evolution generated by a generic Hamiltonian at most bilinear in the mode operators, that is an evolution corresponding to a symplectic transformation in the phase-space [88]. Any mapping, either unitary or completely-positive, transforming Gaussian states into Gaussian states is a Gaussian operation.…”

Section: Quantum Gaussian Statesmentioning

confidence: 99%

Quantifying non-Gaussianity for quantum information

2010

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We address the quantification of non-Gaussianity of states and operations in continuous-variable systems and its use in quantum information. We start by illustrating in details the properties and the relationships of two recently proposed measures of non-Gaussianity based on the Hilbert-Schmidt (HS) distance and the quantum relative entropy (QRE) between the state under examination and a reference Gaussian state. We then evaluate the non-Gaussianities of several families of non-Gaussian quantum states and show that the two measures have the same basic properties and also share the same qualitative behaviour on most of the examples taken into account. However, we also show that they introduce a different relation of order, i.e. they are not strictly monotone each other. We exploit the non-Gaussianity measures for states in order to introduce a measure of non-Gaussianity for quantum operations, to assess Gaussification and de-Gaussification protocols, and to investigate in details the role played by non-Gaussianity in entanglement distillation protocols. Besides, we exploit the QRE-based non-Gaussianity measure to provide new insight on the extremality of Gaussian states for some entropic quantities such as conditional entropy, mutual information and the Holevo bound. We also deal with parameter estimation and present a theorem connecting the QRE non-Gaussianity to the quantum Fisher information. Finally, since evaluation of the QRE non-Gaussianity measure requires the knowledge of the full density matrix, we derive some experimentally friendly lower bounds to non-Gaussianity for some class of states and by considering the possibility to perform on the states only certain efficient or inefficient measurements.

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Section: Quantum Gaussian Statesmentioning

confidence: 99%

Quantifying non-Gaussianity for quantum information

2010

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“…We first note that in view of Williamson's famous diagonalization theorem (see [43]) there exists a symplectic matrix S such that S T MS is diagonal; more precisely…”

Section: The Symplectic Capacity Of An Ellipsoidmentioning

confidence: 99%

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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“…(1). According to Williamson theorem [16], the CM of a N -mode Gaussian state can be always diagonalized by means of a global symplectic transformation (this corresponds to the normal mode decomposition): W σ σW T σ = ν, where W σ ∈ Sp(2N, Ê) and ν = N k=1 diag{ν k , ν k } is the CM corresponding to the tensor product of single-mode thermal states. The quantities {ν k } are the so-called symplectic eigenvalues of the CM σ.…”

Section: Extremal Single-mode Operations and Entanglement Of Purmentioning

confidence: 99%

Geometric characterization of separability and entanglement in pure Gaussian states by single-mode unitary operations

2007

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We present a geometric approach to the characterization of separability and entanglement in pure Gaussian states of an arbitrary number of modes. The analysis is performed adapting to continuous variables a formalism based on single subsystem unitary transformations that has been recently introduced to characterize separability and entanglement in pure states of qubits and qutrits [arXiv:0706.1561]. In analogy with the finite-dimensional case, we demonstrate that the 1 × M bipartite entanglement of a multimode pure Gaussian state can be quantified by the minimum squared Euclidean distance between the state itself and the set of states obtained by transforming it via suitable local symplectic (unitary) operations. This minimum distance, corresponding to a, uniquely determined, extremal local operation, defines a novel entanglement monotone equivalent to the entropy of entanglement, and amenable to direct experimental measurement with linear optical schemes.

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On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems

Cited by 701 publications

References 0 publications

Quantifying non-Gaussianity for quantum information

Quantifying non-Gaussianity for quantum information

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

Geometric characterization of separability and entanglement in pure Gaussian states by single-mode unitary operations

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