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arXiv:1208.2706 · Physical Review A

Genuinely multipartite concurrence of N -qubit X matrices

Seyed Mohammad Hashemi Rafsanjani, Marcus Huber, Curtis J. Broadbent, Joseph H. Eberly

Published Dec 5, 2012• 157 citations
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Abstract

We find an algebraic formula for the $N$-partite concurrence of $N$ qubits in an $X$ matrix. $X$ matrices are density matrices whose only nonzero elements are diagonal or antidiagonal when written in an orthonormal basis. We use our formula to study the dynamics of the $N$-partite entanglement of $N$ remote qubits in generalized $N$-party Greenberger-Horne-Zeilinger (GHZ) states. We study the case in which each qubit interacts with a local amplitude damping channel. It is shown that only one type of GHZ state loses its entanglement in finite time; for the rest, $N$-partite entanglement dies out asymptotically. Algebraic formulas for the entanglement dynamics are given in both cases. We directly confirm that the half-life of the entanglement is proportional to the inverse of $N$. When entanglement vanishes in finite time, the time at which entanglement vanishes can decrease or increase with $N$ depending on the initial state. In the macroscopic limit, this time is independent of the initial entanglement.

DOI
10.1103/physreva.86.062303

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The existence of correlations between the parts of a quantum system on the one hand, and entanglement between them on the other, are different properties. Yet, one intuitively would identify strong N -party correlations with N -party entanglement in an N -partite quantum state. If the local systems are qubits, this intuition is confirmed: The state with the strongest N -party correlations is the Greenberger-Horne-Zeilinger (GHZ) state, which does have genuine multipartite entanglement. However, for high-dimensional local systems the state with strongest N -party correlations may be a tensor product of Bell states, that is, partially separable. We show this by introducing several novel tools for handling the Bloch representation.
In this paper, we first present a simple measure for multiqubit entanglement based on the strategy of bipartite cuts and the measure of negativity. Then, we establish generalized monogamy inequalities and associated partition-dependent residual entanglement (PRE) accounting for arbitrary partitions of a multiqubit system. By virtue of the defined quantities, we investigate the entanglement dynamics of a system of N qubits, either in the Greenberger–Horne–Zeilinger (GHZ)-type state or in the W state, interacting with N independent reservoirs in both Markovian and non-Markovian regimes. We observe entanglement revivals of qubits at instantaneous points of disappearance or after a finite interval of abrupt vanishing due to the memory effect of non-Markovian reservoirs. We also follow the whole entanglement evolution in terms of the PRE to demonstrate the process of transition between the bipartite entanglement of all possible bipartitions and the multipartite entanglement. In particular, we show that the change in time of entanglement formats differs qualitatively for the GHZ-type and W states.
The generalized nnn-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of nnn qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-}1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the nnn-party level.
We consider the entanglement dynamics of N qubits interacting with independent structured reservoirs. The system is initially prepared in the multipartite Greenberger–Horne–Zeilinger (GHZ)-type state. We study the dependence of the entanglement dynamics on both the qubit number N and the purity of initial state, and show that the qubit number N and the purity can control the time of appearance of the N -qubit entanglement sudden death and the entanglement sudden birth of the reservoirs. Also based on the purity and the qubit number N , the revival of N -qubit entanglement does not necessarily indicate the disentanglement of reservoirs, and manifests before, simultaneously or even after the finite-time disentanglement of reservoirs.