The entanglement-sharing properties of an infinite spin-chain are studied when the state of the chain is a pure, translation-invariant state with a matrix-product structure (Klümper A., Schadschneider A. and Zittartz J., J. Phys. A , 24 (1991) L955; Z. Phys. B , 87 (1992) 281; Europhys. Lett. , 24 (1993) 293). We study the entanglement properties of such states by means of their finitely correlated structure (Fannes M., Nachtergaele B. and Werner R. F., Comm. Math. Phys. , 144 (1992) 443; Europhys. Lett. , 10 (1989) 633; J. Phys. A , 24 (1991) L185). These states are recursively constructed by means of an auxiliary density matrix ρ on a matrix algebra ℬ and a completely positive map : ⊗ ℬ → ℬ, where is the spin 2 × 2 matrix algebra. General structural results for the infinite chain are therefore obtained by explicit calculations in (finite) matrix algebras. In particular, we study not only the entanglement shared by nearest-neighbours, but also, differently from previous works (Wootters W. K., Contemp. Math. , 305 (2002) 299) the entanglement shared between connected regions of the spin-chain. This range of possible applications is illustrated and the maximal concurrence = 1/√2 (Coffman V., Kundu J. and Wootters W. K., Phys. Rev. A , 61 (2000) 052306) for the entanglement of connected regions can actually be reached.
We consider translationally invariant states of an infinite one-dimensional chain of qubits or spin-
21 particles. We maximize the entanglement shared by nearest neighbors via a variational approach based on finitely correlated states. We find an upper bound of nearest-neighbor concurrence equal to
C=0.434095 which is
0.09 away from the bound
CW=0.434467 obtained by a completely different procedure. The obtained state-maximizing nearest-neighbor entanglement seems to approximate the maximally entangled mixed states. Further we investigate in detail several other properties of the so-obtained optimal state.
We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.
Bi-partite entanglement in multi-qubit systems cannot be shared freely. The rules of quantum mechanics impose bounds on how multi-qubit systems can be correlated. In this paper we utilize a concept of entangled graphs with weighted edges in order to analyze pure quantum states of multi-qubit systems. Here qubits are represented by vertexes of the graph while the presence of bi-partite entanglement is represented by an edge between corresponding vertexes. The weight of each edge is defined to be the entanglement between the two qubits connected by the edge, as measured by the concurrence. We prove that each entangled graph with entanglement bounded by a specific value of the concurrence can be represented by a pure multi-qubit state. In addition we present a logic network with O(N2) elementary gates that can be used for preparation of the weighted entangled graphs of N qubits.
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.