Lower bound on entanglement of formation for the qubit-qudit system

Gerjuoy, E.

doi:10.1103/physreva.67.052308

Cited by 60 publications

(67 citation statements)

References 11 publications

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“…Even though it has not been known whether the explicit formula of the entanglement of formation for states in 2 ⊗ n quantum system can be computed or not, one can readily compute one of its lower bounds [17,18],…”

Section: Entanglement For the States With Two Parametersmentioning

confidence: 99%

“…Nevertheless, there is no known explicit formula for the entanglement of formation of states in a general quantum system except for states in 2 ⊗ 2 quantum system [2,14], the isotropic states and the Werner states in n ⊗ n quantum system [4,15,16], and states of the specific form [4,5]. For 2 ⊗ n quantum system, only a lower bound on the entanglement of formation is given by decomposing a 2 ⊗ n dimensional Hilbert space into many 2 ⊗ 2 dimensional subspaces [17,18].…”

Section: Introductionmentioning

confidence: 99%

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Entanglement for a two-parameter class of states in 2 nquantum system

Pyo

Lee

2003

J. Phys. A: Math. Gen.

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Abstract. We exhibit a two-parameter class of states ρ (α,γ) , in 2 ⊗ n quantum system for n ≥ 3, which can be obtained from an arbitrary state by means of local quantum operations and classical communication, and which are invariant under all bilateral operations on 2 ⊗ n quantum system. We calculate the negativity of ρ (α,γ) , and a lower bound and a tight upper bound on its entanglement of formation. It follows from this calculation that the entanglement of formation of ρ (α,γ) cannot exceed its negativity.

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Section: Entanglement For the States With Two Parametersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Entanglement for a two-parameter class of states in 2 nquantum system

Pyo

Lee

2003

J. Phys. A: Math. Gen.

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“…and λ ij are the square roots of the four largest eigenvalues of the matrix ρ 1/2 S ij ρ * S ij ρ 1/2 [18]. It does not need numerical optimization for the bound of concurrence of the 2 × M systems.…”

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confidence: 99%

Violation of monogamy inequality for higher-dimensional objects

2007

Phys. Rev. A

102

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Bipartite quantum entanglement for qutrits and higher dimensional objects is considered. We analyze the possibility of violation of monogamy inequality, introduced by Coffman, Kundu, and Wootters, for some systems composed of such objects. An explicit counterexample with a threequtrit totally antisymmetric state is presented. Since three-tangle has been confirmed to be a natural measure of entanglement for qubit systems, our result shows that the three-tangle is no longer a legitimate measure of entanglement for states with three qutrits or higher dimensional objects.PACS numbers: 03.67.Mn, 03.65.Ud Quantification of quantum entanglement plays an important role not only in quantum information processing and quantum computation[1] but also in describing quantum phase transition in various interacting quantum many-body systems [2]. In the last ten years a number of entanglement measures for qubit systems have been studied extensively, in which the well known one with an elegant formula is concurrence derived analytically by Wootters[3], and the entanglement of formation(EOF) [4,5] is a monotonically increasing function of the concurrence. However, at least so far it is believed that there exists a drawback that they are confined into the qubit systems since the used spin-flip is only applicable to qubits [8], Because of which generally only a lower bound of concurrence can be achieved for states composed of qutrits or higher dimensional objects. The seminal paper by Coffman, Kundu, and Wootters[6] provided a basis for the quantification of three-party entanglement by introducing the so called residual tangle, and a general monogamy inequality in the case of n qubits has been proved [7].Since the monogamy inequality has been established, whether it can be generalized to qutrits or higher dimensional objects remains still open. In this Brief Report, we will firstly show that the monogamy inequality can be violated for some quantum composed of qutrits or high dimensional objects, and then offer an explicit example of an antisymmetric state to show this violation. Therefore the main idea here is to show that the monogamy inequality characterized by the concurrence can not be generalized to quantum state apart from qubits. This result gives a caveat when we are studying genuine multipartite entanglement for such states where the residual entanglement or 3-tangle is defined.For completeness we recall the original monogamy inequality. Consider a triple of spin-1/2 particles A, B, and C, and its density matrix is denoted by ρ ABC , the distribution of entanglement among them is constrained by the following inequalitywhere C AB and C AC are the concurrences of the state ρ ABC with traces taken over the particles C, B. C A(BC)is the concurrence of ρ A(BC) with the particles B and C regarded as a single object. In this case the particle A can be viewed as a focus such that the three-tangle can be defined aswhich is independent on the choice of the focus mainly because it is invariant under the permutations of the particles. ...

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“…To find the lower bound of entanglement of formation (EOF) of the qubit-qudit system, E. Gerjuoy [6] first defined d(d − 1)/2 symmetric square matrices S ij , 0 ≤ i ≤ d − 2 and j > i, whose elements all are zero, except for…”

Section: Analyses Of Gefiuoy's Scheme and Partition Of Qubit-qudimentioning

confidence: 99%

“…The quantitative measure of entanglement is one of the main research areas in quantum information theory and quantum computation [2], which have attracted much attention of many researchers. In this sense, many useful measures were developed, such as: concurrence [3][4][5][6][7][8][9][10][11], entanglement of formation (EOF) [12][13][14][15][16][17], geometric measure [18][19][20][21], entanglement witness [22,23], quantum discord [24,25], three-tangle [26], etc. These entanglement measurements are usually defined first for pure states and then are extended to mixed states via the convex roof construction.…”

Section: Introductionmentioning

confidence: 99%

Entanglement of Formation for Qubit-Qudit System Using Partition of Qudit System into a Set of Qubit System

Qiang

Cardoso

2014

Int J Theor Phys

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Gerjuoy [Phys. Rev. A 67, 052308 (2003)] has derived a closed-form lower bound for the entanglement of formation of a mixed qubit-qudit system (qudit system has d levels with d ≥ 3). In this paper, inspired by Gerjuoy's method, we propose a scheme that partitions a qubit-qudit system into d(d−1)/2 qubit-qubit systems, which can be treated by all known methods pertinent to qubit-qubit system. The method is demonstrated by a qubit-qudit system (The levels of qudit are d = 3 and d = 5, respectively).

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Lower bound on entanglement of formation for the qubit-qudit system

Cited by 60 publications

References 11 publications

Entanglement for a two-parameter class of states in 2 nquantum system

Entanglement for a two-parameter class of states in 2 nquantum system

Violation of monogamy inequality for higher-dimensional objects

Entanglement of Formation for Qubit-Qudit System Using Partition of Qudit System into a Set of Qubit System

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