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arXiv:cond-mat/0602268 · Physical Review A

Divergence of the entanglement range in low-dimensional quantum systems

Luigi Amico A, Fabrizio Baroni, Andrea Fubini, Dario Patanè, V. Tognetti, Paola Verrucchi

Published Aug 25, 2006• 116 citations
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Abstract

We study the pairwise entanglement close to separable ground states of a class of one-dimensional quantum spin models. At $T=0$ we find that such ground states separate regions, in the space of the Hamiltonian parameters, which are characterized by qualitatively different types of entanglement, namely parallel and antiparallel entanglement; we further demonstrate that the range of the concurrence diverges while approaching separable ground states, therefore evidencing that such states, with uncorrelated fluctuations, are reached by a long range reshuffling of the entanglement. We generalize our results to the analysis of quantum phase transitions occurring in bosonic and fermionic systems. Finally, the effects of finite temperature are considered: At $T>0$ we evidence the existence of a region where no pairwise entanglement survives, so that entanglement, if present, is genuinely multipartite.

DOI
10.1103/physreva.74.022322

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We study the pairwise entanglement close to separable ground states of a class of one-dimensional quantum spin models. At T=0T=0T=0 we find that such ground states separate regions, in the space of the Hamiltonian parameters, which are characterized by qualitatively different types of entanglement, namely parallel and antiparallel entanglement; we further demonstrate that the range of the concurrence diverges while approaching separable ground states, therefore evidencing that such states, with uncorrelated fluctuations, are reached by a long range reshuffling of the entanglement. We generalize our results to the analysis of quantum phase transitions occurring in bosonic and fermionic systems. Finally, the effects of finite temperature are considered: At T>0T>0T>0 we evidence the existence of a region where no pairwise entanglement survives, so that entanglement, if present, is genuinely multipartite.
We determine the conditions under which general dimer-type spin chains with XYZXYZXYZ couplings of arbitrary range in a general transverse field will exhibit an exactly separable parity-breaking eigenstate. We also provide sufficient conditions which ensure that it will be a ground state. We then examine the exact side limits at separability of the entanglement between any two spins in a finite chain, showing that in the vicinity of separability, the system will loose all signatures of dimerization, with pairwise entanglement approaching infinite range and becoming independent of separation and interaction range. The possibility of a nonuniform exactly separable ground state induced by an alternating field is also shown. As illustration, we examine the behavior of the pairwise entanglement in a finite XYXYXY dimer chain under a uniform as well as alternating field. Related aspects of the magnetization are also discussed.
The interplay of quantum and thermal fluctuations in the vicinity of a quantum critical point characterizes the physics of strongly correlated systems. Here we investigate this interplay from a quantum information perspective presenting the universal phase diagram of the quantum Fisher information at a quantum phase transition. Different regions in the diagram are identified by characteristic scaling laws of the quantum Fisher information with respect to temperature. This feature has immediate consequences on the thermal robustness of quantum coherence and multipartite entanglement. We support the theoretical predictions with the analysis of paradigmatic spin systems showing symmetry-breaking quantum phase transitions and free-fermion models characterized by topological phases. In particular we show that topological systems are characterized by the survival of large multipartite entanglement, reaching the Heisenberg limit at finite temperature.
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We have found that for a wide range of two-qubit Hamiltonians the canonical-ensemble thermal state is entangled in two distinct temperature regions. In most cases the ground state is entangled; however we have also found an example where the ground state is separable and there are still two regions. This demonstrates that the qualitative behavior of entanglement with temperature can be much more complicated than might otherwise have been expected; it is not simply determined by the entanglement of the ground state, even for the simple case of two qubits. Furthermore, we prove a finite bound on the number of possible entangled regions for two qubits, thus showing that arbitrarily many transitions from entanglement to separability are not possible. We also show that only one entangled region is possible for the special case of Hamiltonians without magnetic fields.
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The ground state entanglement in one- and two-dimensional Anderson models are studied with consideration of the long-range correlation effects and using the measures of concurrence and von Neumann entropy. We compare the effects of the long-range power-law correlation for the on-site energies on entanglement with the uncorrelated cases. We demonstrate the existence of the band structure of the entanglement. The intraband and interband jumping phenomena of the entanglement are also reported and explained to as the localization-delocalization transition of the system. We also demonstrated the difference between the results of one- and two-dimensions. Our results show that the correlation of the on-site energies increases the entanglement.