We investigate entanglement properties of N initially disentangled spins, embedded in a ring or d-dimensional lattice. The spins interact via some long-range Ising-type interaction. A description in terms of generalized valence bond solids allows us, for arbitrarily large N, to efficiently calculate reduced density operators of up to ten particles. From these density operators we can compute many quantities, for example the bipartite entanglement S_L between a block of L neighboring spins and the remaining system, lower and upper bounds on the localizable entanglement, and higher order correlation functions. We vary the distance dependence of the interaction and investigate how this variation affects the scaling of the entropy S_L as L grows. We provide a sufficient condition when S_L saturates. In other cases, we find that the entanglement length diverges. For special configurations, we determine S_L analytically for all L in the limit N to infinity.
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.
Abstract We consider one dimensional Ising spin system in a transverse uniform time-dependent magnetic field. The asymptotic behavior of the bipartite entanglements between the terminal spin and each one of the other spins along the chain is investigated and compared at different spin-spin interaction ranges, from nearest neighbor to infinite long range, under the separate action of two different magnetic fields, constant and time-varying. We find that each of the nearest neighbor and next to nearest neighbor bipartite entanglements reach an asymptotic final state that is independent of the initial condition or the variation in the interaction range showing perfect ergodic behavior at quite short interaction ranges. However, the nearest neighbor entanglement maintains this behavior at a slightly longer ranges. The other bipartite entanglements assume a zero value within these interaction ranges. At intermediate short and long interaction ranges, the asymptotic states of all entanglements become strongly dependent on the initial state and the interaction range, deviating from the ergodic behavior observed before. The maximum asymptotic entanglement attainable between a pair of spins takes place at a long interaction range value that increases with the distance between the spins. At the infinite long range interaction, the dynamics of all bipartite entanglements coincide. great care should be taken in constructing both.
We consider the two-spin-subsystem entanglement for eigenstates of the Hamiltonian $H={\ensuremath{\sum}}_{1\ensuremath{\le}j<k\ensuremath{\le}N}{(\frac{1}{{r}_{j,k}})}^{\ensuremath{\alpha}}{\ensuremath{\sigma}}_{j}\ensuremath{\cdot}{\ensuremath{\sigma}}_{k}$ for a ring of
N spin-1/2 particles with associated spin vector operator $(\ensuremath{\hbar}/2){\ensuremath{\sigma}}_{j}$ for the
jth spin. Here
rj,k is the chord distance between sites
j and
k. The case $\ensuremath{\alpha}=2$ corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for $\ensuremath{\alpha}\ensuremath{\ne}2$. Two-spin-subsystem entanglement shows high sensitivity and distinguishes $\ensuremath{\alpha}=2$ from $\ensuremath{\alpha}\ensuremath{\ne}2$. There is no entanglement beyond nearest neighbors for all eigenstates when $\ensuremath{\alpha}=2$. Whereas for $\ensuremath{\alpha}\ensuremath{\ne}2$ one has selective entanglement at any distance for eigenstates of sufficiently high energy in a certain interval of $\ensuremath{\alpha}$ which depends on the energy. The ground state (which is a singlet only for even
N) does not have entanglement beyond nearest neighbors, and the nearest-neighbor entanglement is virtually independent of the range of the interaction controlled by $\ensuremath{\alpha}$.
Abstract The evolution of an N spin-1/2 system with all-range Ising-type interaction is considered. For this system we study the entanglement of one spin with the rest spins. It is shown that the entanglement depends on the number of spins and the initial state. Also, the geometry of the manifold, which contains entangled states, is obtained. For this case we find the dependence of entanglement on the scalar curvature of the manifold and examine it for different numbers of spins in the system. Finally we show that the transverse magnetic field leads to a change in the manifold topology.
The behavior of Ising chains with the spin-spin interaction value λ in a transverse magnetic field of constant intensity ( h = 1) is considered. For a chain of infinite length, exact analytical formulas are obtained for the second central moment (dispersion) of the entropy operator Ŝ = -lnρ with reduced density matrix ρ, which corresponds to a semi-infinite part of the model chain occurring in the ground state. In the vicinity of a critical point λ c = 1, the entanglement entropy fluctuation Δ S (defined as the square root of dispersion) diverges as Δ S ∼ [ln(1/|1 − λ|)] 1/2 . For the known behavior of the entanglement entropy S , this divergence results in that the relative fluctuation δ S = Δ S / S vanishes at the critical point, that is, a state with almost nonfluctuating entanglement is attained.