Disappearance of entanglement: a topological point of view

Zhou, Dong; Joynt, Robert

doi:10.1007/s11128-011-0272-8

Cited by 5 publications

(10 citation statements)

References 67 publications

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“…The use of the entanglement to witness non-Markovianity was first proposed in [107], where the expression (148) was suggested. This proposal has been theoretical addressed for cases of qubits coupled to bosonic environments [115,119,136,146,223,224], for a damped harmonic oscillator [107,196,197], and for random unitary dynamics and classical noise models [102,225]. Experimentally this witness has been analyzed in [172,175,226].…”

Section: Entanglementmentioning

confidence: 99%

Quantum non-Markovianity: characterization, quantification and detection

2014

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We present a comprehensive and up-to-date review of the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of a quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.

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Section: Entanglementmentioning

confidence: 99%

Quantum non-Markovianity: characterization, quantification and detection

2014

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“…It has applications in various different topics, including many body physics [4], local discrimination, quantum computation, condensed matter systems, entanglement witnesses, and the study of quantum channel capacities. The geometric measure of entanglement is nothing but the injective tensor norm itself [8], which appears in the theory of operator algebra [9] and has now become increasingly important in theoretical physics-particularly in quantum channel capacities [10,11,12,13,14,15,16,17,18].A tensor is a multidimensional array [19]. Tensor decompositions originated with Hitchcock in 1927 [20], and the idea of a multiway model is attributed to Cattell in 1944 [21].…”

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confidence: 99%

Geometric Measure of Entanglement and U-Eigenvalues of Tensors

Bai

2014

SIAM J. Matrix Anal. & Appl.

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We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor, and the best complex rank-one approximation. We obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rankone approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.Key words. unitary eigenvalue (U-eigenvalue), Z-eigenvalue, symmetric real tensor, geometric measure of entanglement, the best rank-one approximation 1. Introduction. Entanglement of compound systems is a key resource in quantum information processing. In many practical applications it is of fundamental importance to know whether a state is entangled or not. However, this information is often not sufficient, and it is also required to know how much a state is entangled. A useful tool for quantifying the amount of entanglement of a state is given by the so-called entanglement measures [1, 2].A widely used entanglement measure is provided by the geometric measure of entanglement [3,4,5] that is defined for a pure state. The geometric measure of entanglement was first proposed by Shimony [6] and extended to multipartite systems by Wei and Goldbart [7]. It has applications in various different topics, including many body physics [4], local discrimination, quantum computation, condensed matter systems, entanglement witnesses, and the study of quantum channel capacities. The geometric measure of entanglement is nothing but the injective tensor norm itself [8], which appears in the theory of operator algebra [9] and has now become increasingly important in theoretical physics-particularly in quantum channel capacities [10,11,12,13,14,15,16,17,18].A tensor is a multidimensional array [19]. Tensor decompositions originated with Hitchcock in 1927 [20], and the idea of a multiway model is attributed to Cattell in 1944 [21]. Recently, interest in tensor decompositions has expanded to other fields. Examples include signal processing [22,23], numerical linear algebra [24,25], computer

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“…Positivity requirements on ρ lead to a state space M that is a subset of R 4 N −1 . For given N, M is compact and convex, but its surface has a complicated shape [24,25]. As mentioned in the previous section, GQD introduces a measurement on the all parts of the system and in order to calculate I(Π (1,2) (ρ)) in Eq.…”

Section: Bloch Vector For Generalized Bell Statesmentioning

confidence: 99%

Sudden decoherence transitions for quantum discord

Lim¹,

Joynt²

2014

J. Phys. A: Math. Theor.

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We investigate the disappearance of discord in 2-and multi-qubit systems subject to decohering influences. We formulate the computation of quantum discord and quantum geometric discord in terms of the generalized Bloch vector, which gives useful insights on the time evolution of quantum coherence for the open system, particularly the comparison of entanglement and discord. We show that the analytical calculation of the global geometric discord is NP-hard in the number of qubits, but a similar statement for global entropic discord is more difficult to prove. We present an efficient numerical method to calculating the quantum discord for a certain important class of multipartite states. In agreement with previous work for 2-qubit cases, (Mazzola et al., Phys. Rev. Lett. 104, 200401 (2010)), we find situations in which there is a sudden transition from classical to quantum decoherence characterized by the discord remaining relatively robust (classical decoherence) until a certain point from where it begins to decay quickly whereas the classical correlation decays more slowly (quantum decoherence). However, we find that as the number of qubits increases, the chance of this kind of transition occurring becomes small.

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Disappearance of entanglement: a topological point of view

Cited by 5 publications

References 67 publications

Quantum non-Markovianity: characterization, quantification and detection

Quantum non-Markovianity: characterization, quantification and detection

Geometric Measure of Entanglement and U-Eigenvalues of Tensors

Sudden decoherence transitions for quantum discord

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