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arXiv:1012.4277 · Physical Review A

Entanglement in finite spin rings with noncollinear Ising interaction

Filippo Troiani

Published Feb 24, 2011• 9 citations
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Abstract

We investigate the entanglement properties of finite spin rings, with noncollinear Ising interaction between nearest neighbors. The orientations of the Ising axes are determined either by the spin position within the ring (model $A$) or by the direction of the bond (model $B$). In both cases, the considered spin Hamiltonians have a point-group symmetry instead of the translation invariance that characterizes spin rings with collinear Ising interaction. The ground state of these models exhibit remarkable entanglement properties, resembling Greenberger-Horne-Zeilinger (GHZ)-like states in the absence of an applied magnetic field (model $B$). Besides, the application of a homogeneous magnetic field allows to modify qualitatively the character of the ground-state entanglement, switching from multipartite to pairwise quantum correlations (both models $A$ and $B$).

DOI
10.1103/physreva.83.022324

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Simple physical interactions between spin-1∕21∕21∕2 particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite systems. In particular, we present a detailed investigation of the well-known Ising model of a chain (ring) of spin-1∕21∕21∕2 particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin-1∕21∕21∕2 particles in the chain in the standard (computer) basis, and we investigate quantum entanglement between individual qubits. We analyze bipartite as well as multipartite entanglement in the ground state of the model. In particular, we show that bipartite entanglement between pairs of qubits of the Ising chain (measured in terms of a concurrence) as a function of the parameter $\ensuremath{\lambda}$ has a maximum around the point $\ensuremath{\lambda}=1$, and it monotonically decreases for large values of $\ensuremath{\lambda}$. We prove that in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ this state is locally unitary equivalent to an NNN-partite Greenberger-Horn-Zeilinger state. We also analyze a very specific eigenstate of the Ising Hamiltonian with a zero eigenenergy (we denote this eigenstate as the XXX-state). This XXX-state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset AAA of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of N=2n+1N=2n+1N=2n+1 qubits is maximally entangled with the remaining qubits (set BBB) in the chain. In addition, we prove that by performing a local operation just on the subset BBB, one can transform the XXX-state into a direct product of kkk singlets shared by the parties AAA and BBB. This property of the XXX-state can be utilized for new secure multipartite communication protocols.
We study the entanglement of thermal and ground states in the Heisenberg XX\mathrm{XX}XX qubit rings with a magnetic field. A general result is found that for even-number rings, pairwise entanglement between nearest-neighbor qubits is independent of both the sign of exchange interaction constants and the sign of magnetic fields. As an example we study the entanglement in the four-qubit model and find that the ground state of this model without magnetic fields is shown to be a four-body maximally entangled state measured by the square N-bit concurrence.
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We study translationally invariant rings of qubits with a finite number of sites N, and determine the maximal nearest-neighbor entanglement for a fixed z-component of the total spin. For small numbers of sites we present analytical results. We establish a relation between the maximal nearest-neighbor concurrence and the ground state energy of an XXZ spin model. This connection allows us to calculate the concurrence numerically for N≤24. We point out some interesting finite-size effects. Finally, we generalize our results beyond nearest neighbors.
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We present detailed analytical calculations for an 1D Ising ring of arbitrary number of spin-1/2 particles, in order to reveal entanglement properties of the stationary states. We show that the ground state and specific eigenstates of the Ising Hamiltonian posses remarkable entanglement properties that can reveal new insight into quantum correlations present in the Ising model. This correlations might be exploited in quantum information processing. We propose an intuitive picture of the behaviour of multipartite entanglement and discuss a relation of our results to some aspects of phase transitions in the Ising model.