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arXiv:quant-ph/0006105 · Physical Review A

Multipartite entanglement that is robust against disposal of particles

Wolfgang Dür

Published Jan 16, 2001• 137 citations
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Abstract

We investigate the entanglement properties of multiparticle systems, concentrating on the case where the entanglement is robust against the disposal of particles. Two qubits---belonging to a multipartite system---are entangled in this sense if their reduced density matrix is entangled. We introduce a family of multiqubit states, for which one can choose for any pair of qubits independently whether they should be entangled or not as well as the relative strength of the entanglement, thus providing the possibility to construct all kinds of ``entanglement molecules.'' For some particular configurations, we also give the maximal amount of entanglement achievable.

DOI
10.1103/physreva.63.020303

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We investigate the entanglement properties of multiparticle systems, concentrating on the case where the entanglement is robust against the disposal of particles. Two qubits---belonging to a multipartite system---are entangled in this sense if their reduced density matrix is entangled. We introduce a family of multiqubit states, for which one can choose for any pair of qubits independently whether they should be entangled or not as well as the relative strength of the entanglement, thus providing the possibility to construct all kinds of ``entanglement molecules.'' For some particular configurations, we also give the maximal amount of entanglement achievable.
We propose a scheme to prepare a certain kind of N-atom entangled state that allows us to construct some possible types of entanglement molecules via cavity QED. The entanglement properties of entanglement molecules \psi(N)>(alpha) are studied with respect to bipartite entanglement that is robust against the disposal of particles and are compared with entanglement molecules p, introduced in Dur (2001 Phys. Rev. A 63 020303). We also give the maximal amount of entanglement achievable for two particular situations in two possible configurations. Meanwhile, we investigate the entanglement properties of entanglement molecules \psi(N)>(alpha) in terms of local measurement using the maximum connectedness and persistency and compare them with other kinds of N-atom entangled states such as \GHZ>, \W-N> and \phi(N)>. We show that the maximal value N - 1 of the persistency of the state \psi(N)>(alpha) corresponds to the case that all atoms are pairwise entangled. If any pair of atoms rho(ij) is disentangled, the entanglement of the state \psi(N)>(alpha) is very easy to destroy by a single local measurement.
We discuss several aspects of multiparticle mixed-state entanglement and its experimental detection. First we consider entanglement between two particles which is robust against disposals of other particles. To completely detect these kinds of entanglement, full knowledge of the multiparticle density matrix (or of all reduced density matrices) is required. Then we review the relation of the separability properties of l -partite splittings of a state ρ to its multipartite entanglement properties. We show that it suffices to determine the diagonal matrix elements of ρ in a certain basis in order to detect multiparticle entanglement properties of ρ. We apply these observations to analyse two recent experiments, where multiparticle entangled states of 3 (4) particles were produced. Finally, we focus on bound entangled states (non-separable, non-distillable states) and show that they can be activated by joint actions of the parties. We also provide several examples which show the activation of bound entanglement with bound entanglement.
When some of the parties of a multipartite entangled pure state are lost, the question arises whether the residual mixed state is also entangled, in which case the initial entangled pure state is said to be robust against particle loss. In this paper, we investigate this entanglement robustness for NNN-qubit pure states. We identify exhaustively all entangled states that are fragile, i.e., not robust, with respect to the loss of any single qubit of the system. We also study the entanglement robustness properties of symmetric states and put these properties in the perspective of the classification of states with respect to stochastic local operations assisted with classic communication (SLOCC classification).
Given a multipartite quantum system that consists of two-level particles (qubits), one may or may not have access to all the subsystems. What can we know about the entanglement of the multiqubit system and residual correlations beyond two-tangle if we have access only to two-qubits at a time? Algebraic analysis of two-qubit states yields monogamy constraints on distribution of entanglement between sub-systems of an N-qubit state and criterion to determine if the state has multipartite entanglement. Monogamy constraints, reported in this letter, are relations between well known entanglement measures such as one-tangle, two-tangles and three-tangles of an N-qubit pure state.
Beyond the simplest case of bipartite qubits, the composite Hilbert space of multipartite systems is largely unexplored. In order to explore such systems, it is important to derive analytic expressions for parameters which characterize the system's state space. Two such parameters are the degree of genuine multipartite entanglement and the degree of mixedness of the system's state. We explore these two parameters for an N-qubit system whose density matrix has an X form. We derive the class of states that has the maximum amount of genuine multipartite entanglement for a given amount of mixedness. We compare our results with the existing results for the N = 2 case. The critical amount of mixedness above which no N-qubit X-state possesses genuine multipartite entanglement is derived. It is found that as N increases, states with higher mixedness can still be entangled.