Convex-roof extended negativity as an entanglement measure for bipartite quantum systems

Lee, Soojoon; Pyo, Dong; Oh, Sangwon; Kim, Jaewan

doi:10.1103/physreva.68.062304

Cited by 159 publications

(120 citation statements)

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“…Each Hilbert direction-subspace has a dimension growing with the number of time steps, so we use the generalization of the negativity for higher-dimensional systems (so as to have 0 N 1) [30]. We have calculated N (3) in AQW (3) , with the walker starting at the origin and initial coin state col(1,i)/ √ 2, for a number of time steps t up to 10, obtaining the plot in Fig.…”

Section: Generation Of Multipartite Entanglementmentioning

confidence: 99%

N-dimensional alternate coined quantum walks from a dispersion-relation perspective

et al. 2013

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We suggest an alternative definition of N -dimensional coined quantum walk by generalizing a recent proposal [Di Franco et al., Phys. Rev. Lett. 106, 080502 (2011)]. This N -dimensional alternate quantum walk, AQW (N) , in contrast with the standard definition of the N -dimensional quantum walk, QW (N) , requires only a coin qubit. We discuss the quantum diffusion properties of AQW (2) and AQW (3) by analyzing their dispersion relations that reveal, in particular, the existence of diabolical points. This allows us to highlight interesting similarities with other well-known physical phenomena. We also demonstrate that AQW (3) generates considerable genuine multipartite entanglement. Finally, we discuss the implementability of AQW (N) In both its standard forms, the coined [1] and the continuous one [2], quantum walk is the quantum version of a classical random process, described by the diffusion and the telegrapher's equations, respectively [3]. In the coined quantum walk-the process we consider here-there is a system (the walker) that undergoes a conditional displacement, to the right or the left, depending on the output of a coin throw, as in the random walk. But differently from its classical counterpart, here both coin and walker are quantum in nature. The onedimensional coined quantum walk-QW (1) for short-has been studied from many different perspectives, especially from the quantum computational point of view [4]. In the last few years, quantum walks have also received increasing experimental attention [5][6][7], including cases with more than one particle [8].The situation is quite different when dealing with Ndimensional quantum walks, QW (N) for short. They were first discussed by Mackay et al., who introduced them in complete analogy with QW (1) [9] (see also Ref. [10]). As defined in Ref. [9], QW (N) requires the use of a 2 N -dimensional qudit as coin, as well as a coin operator represented by a 2 N × 2 N unitary matrix. This introduces increasing complexity in the process as N grows, especially from the experimental viewpoint [11,12], but also from the theoretical one [13,14]. However, Di Franco et al. [15] have recently proposed an alternative twodimensional quantum walk, namely, the alternate quantum walk-AQW for short-that is simpler than the standard one. In AQW the coin is a single qubit, as in QW (1) , and each time step is divided into two halves: in the first one the coin is thrown (i.e., a Hadamard transformation is applied on the coin qubit) and the conditional displacement along x is performed; then, in the second half of the time step, the coin is thrown again and the conditional displacement along y is performed. Hence, in AQW, the four-dimensional qudit of QW (2) is replaced by a single qubit, the price paid for that being to double the number of substeps per single time step. Quite unexpectedly, AQW reproduces the same spatial probability distributions of QW (2) when the Grover coin is used -Grover-QW (2) for short -for a set of particular initial conditions, precisely those for ...

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Section: Generation Of Multipartite Entanglementmentioning

confidence: 99%

N-dimensional alternate coined quantum walks from a dispersion-relation perspective

et al. 2013

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“…The bound can be made even better if it is much easier to compute a convex-roof extended entanglement measure [22,30] ρ TA co or R(ρ) co , than the EOF. The extended measures in our case are defined by…”

Section: Examplementioning

confidence: 99%

“…They have been studied and calculated for some special class of states in [22,30]. Defining Λ = max( ρ TA co , R(ρ) co ), one finds that the result of the Theorem is still valid, since the last inequality in Eq.…”

Section: Examplementioning

confidence: 99%

Entanglement of Formation of Bipartite Quantum States

2005

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We give an explicit tight lower bound for the entanglement of formation for arbitrary bipartite mixed states by using the convex hull construction of a certain function. This is achieved by revealing a novel connection among the entanglement of formation, the well-known Peres-Horodecki and realignment criteria. The bound gives a quite simple and efficiently computable way to evaluate quantitatively the degree of entanglement for any bipartite quantum state.PACS numbers: 03.67. Mn, 03.65.Ud, 89.70.+c Quantum entangled states are used as key resources in quantum information processing and communication, such as in quantum cryptography, quantum teleportation, dense coding, error correction and quantum computation [1]. A fundamental problem in quantum information theory is how to quantify the degree of entanglement in a practical and operational way [2,3]. One of the most meaningful and physically motivated measures is the entanglement of formation (EOF) [2,3], which quantifies the minimal cost needed to prepare a certain quantum state in terms of EPR pairs. Related to the EOF the behavior of entanglement has recently been shown to play important roles in quantum phase transition for various interacting quantum many-body systems [4] and may significantly affect macroscopic properties of solids [5]. Moreover, it has been shown that there is a remarkable connection between entanglement and the capacity of quantum channels [6]. A quantitative evaluation of EOF is thus of great significance both theoretically and experimentally.Considerable efforts have been spent on deriving EOF or its lower bound through analytical and numerical approaches, for some limited sets of mixed states [7,8,9,10,11,12,13,14,15,16,17]. Among them, the most noteworthy results are an elegant analytical formula for two qubits [7,8] [18,19] in giving analytic lower bounds [that can be optimized further numerically [18]] for the concurrence, which permits to furnish a lower bound of EOF for a generic mixed state. However, this lower bound is not explicit except for the case of 2 ⊗ n systems [19]. For low dimensional systems, numerical methods [10] can be used to estimate EOF. Nevertheless, they are generally time-consuming and often not very efficient. The notorious difficulty of evaluation for EOF is due to the complexity to solve a high dimensional optimization problem, which becomes a formidable task, as the dimensionality of the Hilbert space grows.In this Letter, we present the first analytical calculation of a tight lower bound of EOF for arbitrary bipartite quantum states. An explicit expression for the bound is obtained from the convex hull of a simple function, based on a known result in [9]. This is achieved by establishing a key connection between EOF and two strong separability criteria, the Peres-Horodecki criterion [20,21] and the realignment criterion [22,23]. The bound is shown to be exact for some special states such as isotropic states [9,24] and permits to provide EOF estimations for many bound entangled states (BES). It provi...

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“…It satisfies all the criteria needed in the measure and it has been proved that the negativity an entanglement monotone and therefore it is a good measure of entanglement [22,23]. For mixed states, it gives the same results obtained from the other measures as concurrence [24]. Also, the negativity and the relative entropy of entanglement and lead to upper bounds on the entanglement [25].…”

Section: Entanglement Dynamicsmentioning

confidence: 55%

Information Loss in Local Dissipation Environments

Metwally¹

2010

Int J Theor Phys

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The sensitivity of entanglement to the thermal and squeezed reservoirs' parameters is investigated regarding entanglement decay and what is called sudden-death of entanglement, for a system of two qubit pairs. The dynamics of information is investigated by means of the information disturbance and exchange information. We show that for squeezed reservoir, we can keep both of the entanglement and information survival for a long time.However the sudden death of information is depicted clearly when the entangled qubit system interacts with thermal reservoir. By controlling the environmental parameters, one can reduces the lose of the encrypted information.

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Convex-roof extended negativity as an entanglement measure for bipartite quantum systems

Cited by 159 publications

References 26 publications

N-dimensional alternate coined quantum walks from a dispersion-relation perspective

N-dimensional alternate coined quantum walks from a dispersion-relation perspective

Entanglement of Formation of Bipartite Quantum States

Information Loss in Local Dissipation Environments

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