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arXiv:quant-ph/0502160 · New Journal of Physics

Multipartite entanglement in spin chains

Otfried Gühne, Géza Tóth, Hans J. Briegel

Published Nov 4, 2005• 243 citations
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Abstract

We investigate the presence of multipartite entanglement in macroscopic spin chains. We discuss the Heisenberg and the XY model and derive bounds on the internal energy for systems without multipartite entanglement. Based on this we show that in thermal equilibrium the above-mentioned spin systems contain genuine multipartite entanglement, even at finite modest temperatures.

DOI
10.1088/1367-2630/7/1/229

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We investigate the relation between the entanglement properties of a quantum state and its energy for macroscopic spin models. To this aim, we develop a general method to compute energy bounds for states without certain forms of multipartite entanglement. Violation of these bounds implies the presence of these types of multipartite entanglement. As examples, we investigate the Heisenberg model in different dimensions, the Ising model and the XXXXXX model in the presence of a magnetic field. Finally, by studying the Heisenberg model on a triangular lattice, we demonstrate that our techniques can be applied also to frustrated systems.
We analyze the XYXYXY model characterized by an anisotropy $\ensuremath{\gamma}$ in an external magnetic field, hhh, with respect to its genuine multipartite entanglement content (in the thermodynamic and finite-size case). Despite its simplicity we show that the quantity---detecting genuine multipartite entanglement through permutation operators and being a lower bound on measures---witnesses the presence of genuine multipartite entanglement for nearly all values of $\ensuremath{\gamma}$ and hhh. We further show that the phase transition and scaling properties are fully characterized by this multipartite quantity. Consequently, we provide a useful toolbox for other condensed-matter systems, where bipartite entanglement measures are known to fail.
We study the occurrence of multipartite entanglement in spin chains. We show that certain genuine multipartite entangled states, namely WWW states, can be obtained as ground states of simple XX type ferromagnetic spin chains in a transverse magnetic field, for any number of sites. Moreover, multipartite entanglement is proven to exist even at finite temperatures. A transition from a product state to a multipartite entangled state occurs when decreasing the magnetic field to a critical value. Adiabatic passage through this point can thus lead to the generation of multipartite entanglement.
Abstract Multipartite entanglement (ME) in the XY spin-1/2 chain is investigated by using negativity as a quantitative measure. The results show that ME persists over longer distances than bipartite entanglement, with its range and strength dependent on the anisotropy parameter γ, external magnetic field h, and the size of spin blocks. Notably, ME could serve as a sensitive probe for quantum phase transitions, due to the divergent behavior in its first derivative at the critical point h = 1. In essence, ME may primarily derive from collective effects rather than pairwise spin-spin entanglement, particularly in disjoint spin configurations. At finite temperatures and γ= 1, the thermal suppression of ME follows a hierarchical pattern, and such suppression is further modulated by ℎ and the specific block geometries. This behavior challenges the conventional conjecture of a simplistic monotonic decay.
Genuine multipartite entanglement has been found in some spin chain systems. However, genuine multipartite nonlocality is much rarer than genuine multipartite entanglement. Here we present genuine multipartite nonlocality in a spin chain system. After introducing the definition of genuine multipartite nonlocality and a multipartite Bell-type inequality, we construct a group of joint measurements for the inequality in a one-dimensional ferromagnetic NNN-qubit chain with nearest-neighbor XXZXXZXXZ interaction, and many violations to the inequality have been found. The violations do indicate that genuine multipartite nonlocality exists in this ferromagnetic spin-1/2 chain system. Last but not least, we also calculate genuine multipartite entanglement concurrence in the same spin chain to demonstrate the difference and relationship between genuine multipartite nonlocality and genuine multipartite entanglement.