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arXiv:0802.1975 · Physical Review A

Genuine tripartite entanglement monotone of (2⊗2⊗n) -dimensional systems

Chang‐shui Yu, Ling Zhou, He‐Shan Song

Published Feb 13, 2008• 12 citations
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Abstract

A genuine tripartite entanglement monotone is presented for $(2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n)$-dimensional tripartite pure states, obtained by introducing an entanglement measure for bipartite pure states. As an application, we consider the genuine tripartite entanglement of the ground state of the exactly solvable isotropic spin-$1/2$ chain with three-spin interaction. It is shown that the singular behavior of the genuine tripartite entanglement exactly signals a quantum phase transition.

DOI
10.1103/physreva.77.022313

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A genuine tripartite entanglement monotone is presented for (2\ensuremath⨂2\ensuremath⨂n)(2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n)(2\ensuremath⨂2\ensuremath⨂n)-dimensional tripartite pure states, obtained by introducing an entanglement measure for bipartite pure states. As an application, we consider the genuine tripartite entanglement of the ground state of the exactly solvable isotropic spin-1/21/21/2 chain with three-spin interaction. It is shown that the singular behavior of the genuine tripartite entanglement exactly signals a quantum phase transition.
A new formulation called as entanglement measure for simplification, is presented to characterize genuine tripartite entanglement of (2×2×n)−(2\times 2\times n)-(2×2×n)−dimensional quantum pure states. The formulation shows that the genuine tripartite entanglement can be described only on the basis of the local (2×2)−(2\times 2)-(2×2)−dimensional reduced density matrix. In particular, the two exactly solvable models of spin system studied by Yang (Phys. Rev. A \textbf{71}, 030302(R) (2005)) is reconsidered by employing the entanglement measure. The results show that a discontinuity in the first derivative of the entanglement measure or in the entanglement measure itself of the ground state just corresponds to the existence of quantum phase transition, which is obviously prior to concurrence. Hence, the given entanglement measure may become a new alternate candidate to help study the connection between quantum entanglement and quantum phase transitions.
In this paper, we present a new approach to study genuine tripartite entanglement existing in (2×2×n)−(2\times 2\times n)-(2×2×n)−dimensional quantum pure states. By utilizing the approach, we introduce a particular quantity to measure genuine tripartite entanglement. The quantity is shown to be an entanglement monotone in 2-dimensional subsystems (semi-monotone) and reaches zero for separable states and (2×2×2)−(2\times 2\times 2)-(2×2×2)−dimensional WWW states, hence is a good criterion to characterize genuine tripartite entanglement. Furthermore, the formulation for pure states can be conveniently extended to the case of mixed states by utilizing the kronecker product approximation technique. As applications, we give the analytic approximation for weakly mixed states, and study the genuine tripartite entanglement of two given weakly mixed states.
A new simplified formula is presented to characterize genuine tripartite entanglement of (2 ⊗ 2 ⊗ n )-dimensional quantum pure states. The formula turns out equivalent to that given in ( Quant. Inf. Comp. 7 (7) 584 (2007)), hence it also shows that the genuine tripartite entanglement can be described only on the basis of the local (2 ⊗ 2)-dimensional reduced density matrix. In particular, the two exactly solvable models of spin system studied by Yang ( Phys. Rev. A 71 030302(R) (2005)) are reconsidered by employing the formula. The results show that a discontinuity in the first derivative of the formula or in the formula itself of the ground state just corresponds to the existence of quantum phase transition, which is obviously different from the concurrence.
We evaluate and analyze the exact value of a measure for local genuine tripartite entanglement in the one-dimensional cluster-Ising model for spin- particles. This model is attractive since cluster states are considered to be relevant sources for applying quantum algorithms and the model is experimentally feasible. Whereas bipartite entanglement is identically vanishing, we find that genuine tripartite entanglement is non zero in the anti-ferromagnetic phase and also in the cluster phase well before the critical point. We prove that the measure of local genuine tripartite entanglement captures all the properties of the symmetry-protected topological quantum phase transition. Remarkably, we find that the amount of genuine tripartite entanglement is independent of whether the ground states satisfy or break the symmetries of the Hamiltonian. We provide also strong evidences that local genuine tripartite entanglement represents the unique non-vanishing genuine multipartite entanglement.
An interesting monogamy equation with the form of the Pythagorean theorem is found for $2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n$-dimensional pure states, which reveals the relation among bipartite concurrence, concurrence of assistance, and genuine tripartite entanglement. At the same time, a genuine tripartite entanglement monotone as a generalization of 3-tangle is naturally obtained for $(2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n)$-dimensional pure states in terms of a distinct idea. For mixed states, the monogamy equation is reduced to a monogamy inequality. Both results for tripartite quantum states can be employed to multipartite quantum states.
Usually, the three-tangle of a tripartite pure state of qubits can be directly measured with the simultaneous preparation of a not-less-than fourfold copy of the state. We show that the exact genuine tripartite entanglement for ($2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n$)-dimensional pure quantum states can be measured in a similar manner, provided that only two simultaneous copies of the state are available. Lower bounds are also proposed for more convenient experimental operations. As an example, a comprehensive demonstration of the scheme is provided for the three-tangle of a three-qubit state.