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arXiv:0806.0779 · Physical Review A

Robustness of highly entangled multiqubit states under decoherence

Antoni Borràs, A. P. Majtey, Angel Ricardo Plastino, M. Casas, Angel Ricardo Plastino

Published Feb 9, 2009• 67 citations
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Abstract

We investigate the decay of entanglement, due to decoherence, of multiqubit systems that are initially prepared in highly (in some cases maximally) entangled states. We assume that during the decoherence processes each qubit of the system interacts with its own independent environment. We determine, for systems with a small number of qubits and for various decoherence channels, the initial states exhibiting the most robust entanglement. We also consider a restricted version of this robustness-optimization problem that only involves states equivalent, under local unitary transformations, to the Greenberger-Horne-Zeilinger $|\text{GHZ}⟩$ state.

DOI
10.1103/physreva.79.022108

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We investigate the decay of entanglement, due to decoherence, of multiqubit systems that are initially prepared in highly (in some cases maximally) entangled states. We assume that during the decoherence processes each qubit of the system interacts with its own independent environment. We determine, for systems with a small number of qubits and for various decoherence channels, the initial states exhibiting the most robust entanglement. We also consider a restricted version of this robustness-optimization problem that only involves states equivalent, under local unitary transformations, to the Greenberger-Horne-Zeilinger ∣GHZ⟩|\text{GHZ}⟩∣GHZ⟩ state.
We study the robustness of multiparty entanglement under local decoherence, modeled by partially depolarizing channels acting independently on each subsystem. Surprisingly, we find that the robustness of (GHZ) entanglement increases with the number of qubits. An n-qubit Greenberger-Horne-Zeilinger (GHZ) state can stand more than 55% local depolarization in the limit $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}.$ GHZ states are more robust than other generic states of three and four qubits. We also study spin-squeezed states in the limit $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}$ and find that they too can stand considerable local depolarization.
In practice, the interaction between quantum system and environment will lead to the decoherence of entanglement. By quantifying the entanglement through the correlation tensor, we systematically investigate the entanglement robustness of two-qubit states under the four types of noisy channels and corresponding correlated channels. We use an analytical method to find all of the most robust and fragile states under the noisy channels. The puzzling phenomenon that the most robust and fragile states have the same entanglement under noisy channels has been investigated. Moreover, we compare the entanglement robustness of a three-qubit Greenberger-Horne-Zeilinger state with a WWW state. Our method shows the behavior of entanglement under the noisy channels and characterizes the entanglement robustness of the multiqubit system analytically.
We study the robustness of the Greenberger-Horne-Zeilinger (``cat'') class of multipartite states under decoherence. The noise model is described by a general completely positive map for qubits independently coupled to the environment. In particular, the robustness of NNN-party entanglement is studied in the large NNN limit when (a) the number of spatially separated subsystems is fixed but the size of each subsystem becomes large and (b) the size of the subsystems is fixed while their number becomes arbitrarily large. We obtain conditions for entanglement in these two cases. Among our other results, we show that the parity of an entangled state (i.e., whether it contains an even or odd number of qubits) can lead to qualitatively different robustness of entanglement under certain conditions.
We investigate the robustness of multiparty nonlocality under local decoherence, acting independently and equally on each subsystem. To be specific, we consider an NNN-qubit Greenberger-Horne-Zeilinger (GHZ) state under a depolarization, dephasing, or dissipation channel, and examine nonlocality by testing violation of the Mermin-Klyshko inequality, which is one of Bell's inequalities for multiqubit systems. The results show that the robustness of nonlocality increases with the number of qubits, and that the nonlocality of an NNN-qubit GHZ state with even NNN is extremely persistent against dephasing.
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.
We investigate the decay of the global entanglement, due to decoherence, of multiqubit systems interacting with a reservoir in a non Markovian regime. We assume that during the decoherence process each qubit of the system interacts with its own, independent environment. Most previous works on this problem focused on particular initial states or families of initial states amenable of analytical treatment. Here we determine numerically the typical, average behaviour of the system corresponding to random initial pure states uniformly distributed (in the whole Hilbert space of nnn-qubit pure states) according to the Haar measure. We study systems consisting of 3, 4, 5, and 6 qubits. In each case we consider also the entanglement dynamics corresponding to important particular initial states, such as the GHZ states or multiqubit states maximizing the global entanglement, and determine in which cases any of these states is representative of the average behaviour associated with general initial states.