Introduction to the Basics of Entanglement Theory  in Continuous-Variable Systems

Eisert, Jens; Plenio, Martin B.

doi:10.1142/s0219749903000371

Cited by 340 publications

(448 citation statements)

References 80 publications

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“…We consider a continuous-variable (CV) system consisting of N canonical bosonic modes, associated with an infinite-dimensional Hilbert space, tensor product of the N single-mode Fock spaces [11,12,14]. Unitary operations which are at most quadratic in the canonical operators, amount to symplectic transformations in phase space.…”

Section: Phase-space Description Of Gaussian States and Single-momentioning

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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Section: Phase-space Description Of Gaussian States and Single-momentioning

confidence: 99%

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

2007

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“…The symplectic matrix corresponds to an encoding unitary acting on the unencoded quantum state [19,20]. This correspondence results from the Stone-von Neumann Theorem and unifies the Schrödinger and Heisenberg pictures for quantum error correction [30].…”

mentioning

confidence: 87%

Encoding one logical qubit into six physical qubits

et al. 2008

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We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum errorcorrecting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a CalderbankShor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver's half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender's side. We discuss the advantages and disadvantages for each of the two codes.

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“…The new covariance matrixṼ is connected to V by the following relation [40,41],Ṽ = SV S † , (A. 4) which implies that…”

Section: A Obtaining the Standard Formṽmentioning

confidence: 99%

Quantification of continuous variable entanglement with only two types of simple measurements

Rigolin¹,

Oliveira²

2008

Annals of Physics

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Here we propose an experimental set-up in which it is possible to obtain the entanglement of a two-mode Gaussian state, be it pure or mixed, using only simple linear optical measurement devices. After a proper unitary manipulation of the two-mode Gaussian state only number and purity measurements of just one of the modes suffice to give us a complete and exact knowledge of the state's entanglement.

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Introduction to the Basics of Entanglement Theory in Continuous-Variable Systems

Cited by 340 publications

References 80 publications

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

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

Encoding one logical qubit into six physical qubits

Quantification of continuous variable entanglement with only two types of simple measurements

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