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arXiv:quant-ph/0608236 · Physical Review A

Robustness of multiparty nonlocality to local decoherence

Sung Soon Jang, Yong Wook Cheong, Jaewan Kim, Hai-Woong Lee

Published Dec 27, 2006• 16 citations
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Abstract

We investigate the robustness of multiparty nonlocality under local decoherence, acting independently and equally on each subsystem. To be specific, we consider an $N$-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 $N$-qubit GHZ state with even $N$ is extremely persistent against dephasing.

DOI
10.1103/physreva.74.062112

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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.
We study the nonlocal properties of two-qubit maximally entangled and NNN-qubit Greenberger-Horne-Zeilinger states under local decoherence. We show that the (non)resilience of entanglement under local depolarization or dephasing is not necessarily equivalent to the (non)resilience of Bell-inequality violations. Apart from entanglement and Bell-inequality violations, we consider also nonlocality as quantified by the nonlocal content of correlations and provide several examples of anomalous behaviors, both in the bipartite and multipartite cases. In addition, we study the practical implications of these anomalies on the usefulness of noisy Greenberger-Horne-Zeilinger states as resources for nonlocality-based physical protocols given by communication complexity problems.
The Clauser-Horne-Shimony-Holt inequality was originally proposed as a Bell inequality to detect nonlocality in bipartite systems. However, it can also be used to certify the nonlocality of multipartite quantum states. We apply this to study the nonlocality of multipartite Greenberger-Horne-Zeilinger (GHZ), W, and graph states under local decoherence processes. We derive lower bounds on the critical local-noise strength tolerated by the states before becoming local. In addition, for the whole noisy dynamics, we derive lower bounds on the corresponding nonlocal content for the three classes of states. All the bounds presented can be calculated efficiently and, in some cases, provide significantly tighter estimates than with any other known method. For example, they reveal that NNN-qubit GHZ states undergoing local dephasing are, for all NNN, nonlocal throughout all the dephasing dynamics.
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.
The N-qubit states of the W class, for N>10,N>10,N>10, lead to more robust (against noise admixture) violations of local realism, than the Greenberger-Horne-Zeilinger (GHZ) states. These violations are most pronounced for correlations for a pair of qubits, conditioned on specific measurement results for the remaining $N\ensuremath{-}2$ qubits. The considerations provide us with a qualitative difference between the W state and GHZ state in the situation when they are separately sent via depolarizing channels. For sufficiently high amount of noise in the depolarizing channel, the GHZ states cannot produce a distillable state between two qubits, whereas the W states can still produce a distillable state in a similar situation.
We investigate genuine multiqubit nonlocality in families of entangled 3- and 4-qubit pure states by analyzing a Bell-type inequality that is violated only if all qubits are nonlocally correlated. We present detailed numerical studies of the relationship between entanglement and violation of the Svetlichny Bell-type inequality in an experimentally accessible set of 3-qubit pure states, and identify the special nonlocality property of the maximal slice states in the space of all 3-qubit pure states. We also analyze nonlocal correlations in 3-qubit generalized Greenberger–Horne–Zeilinger (GHZ) states and extend our analysis to the case of 4-qubit generalized GHZ states. We show that like the 3-qubit case, some 4-qubit generalized GHZ states do not violate a Bell inequality that tests for genuine 4-qubit nonlocality. Furthermore, the location of the boundary between the states that do violate the inequality and those that do not is the same for the 3- and 4-qubit generalized GHZ states.