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arXiv:0903.0717 · Physical Review A

Decay of multiqudit entanglement

Zhao Liu, Heng Fan

Published Jun 25, 2009• 37 citations
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Abstract

We investigate the decay of entanglement of a generalized $N$-qudit Greenberger-Horne-Zeilinger state with each qudit passing through independently in a quantum noisy channel. By studying the time at which the entanglement completely vanishes and the time at which the entanglement becomes arbitrarily small, we try to find how the robustness of entanglement is influenced by the dimension $d$ and the number of particles $N$.

DOI
10.1103/physreva.79.064305

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We investigate the disentanglement dynamics of generalized multiqubit Greenberger-Horne-Zeilinger-type states $\ensuremath{\mid}{\ensuremath{\Psi}}_{n,N}⟩=\ensuremath{\alpha}{\ensuremath{\mid}1⟩}^{\ensuremath{\bigotimes}n}{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}(N\ensuremath{-}n)}+\ensuremath{\beta}{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}n}{\ensuremath{\mid}1⟩}^{\ensuremath{\bigotimes}(N\ensuremath{-}n)}$ (with $1\ensuremath{\leqslant}n<N$) in a local decoherence environment. By adopting the time after which all the bipartite entanglements vanish or become negligibly small as the criterion of entanglement robustness, we show that states $\ensuremath{\mid}{\ensuremath{\Psi}}_{n,N}⟩$ are more robust than states $\ensuremath{\mid}{\ensuremath{\Psi}}_{N}⟩=\ensuremath{\alpha}{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}N}+\ensuremath{\beta}{\ensuremath{\mid}1⟩}^{\ensuremath{\bigotimes}N}$ in all respects. Since the two states $\ensuremath{\mid}{\ensuremath{\Psi}}_{N}⟩$ and $\ensuremath{\mid}{\ensuremath{\Psi}}_{n,N}⟩$ can be transformed into each other by local operations, we confirm that entanglement robustness can be enhanced by local operations though the amount of entanglement cannot.
How much noise can a given quantum state tolerate without losing its entanglement? For qudits of arbitrary dimension, I investigate this question for two noise models: Global white noise, where a depolarizing channel is applied to all qudits simultaneously, and local white noise, where a single qudit depolarizing channel is applied to every qudit individually. Using a unitary generalization of the Pauli group, I derive noise thresholds for stabilizer states, with an emphasis on graph states, and compare different entanglement criteria. The PPT and reduction criteria generally provide high noise thresholds, however, it is difficult to apply them in the case of local white noise. Entanglement criteria based on so-called sector lengths, on the other hand, provide coarse noise thresholds for both noise models. The only thing one has to know about a state to compute this threshold is the number of its full-weight stabilizers. In the special case of qubit graph states, I relate this question to a graph-theoretical puzzle and solve it for four important families of states. For Greenberger-Horne-Zeilinger states under local white noise, I obtain for the first time a noise threshold below which so-called semiseparability is ruled out.
We present a method to determine the decay of multiparticle quantum correlations as quantified by the geometric measure of entanglement under the influence of decoherence. With this, we compare the robustness of entanglement in Greenberger-Horne-Zeilinger (GHZ), cluster, WWW, and Dicke states of four qubits and show that the Dicke state is the most robust. Finally, we determine the geometric measure analytically for decaying GHZ and cluster states of an arbitrary number of qubits.