Quantum-information approach to the Ising model: Entanglement in chains of qubits

Abstract

Simple physical interactions between spin-$1∕2$ 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-$1∕2$ particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin-$1∕2$ 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 $N$-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 $X$-state). This $X$-state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset $A$ of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of $N=2n+1$ qubits is maximally entangled with the remaining qubits (set $B$) in the chain. In addition, we prove that by performing a local operation just on the subset $B$, one can transform the $X$-state into a direct product of $k$ singlets shared by the parties $A$ and $B$. This property of the $X$-state can be utilized for new secure multipartite communication protocols.