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.
We investigate multipartite entanglement and a quantum phase transition for spin chain systems by making use of the density-matrix renormalization-group method in the form of matrix-product states. It is found that multipartite entanglement changes dramatically near the critical region and the residual-to-global entanglement ratio reaches its maximum at the critical point of the quantum phase transition. These results indicate that multipartite entanglement as well as the residual-to-global entanglement ratio can serve as good indicators to detect a quantum phase transition. We also demonstrate a linear relation between the next-nearest-neighbor interaction strength and the critical magnetic field for an
XYZ spin chain by using multipartite entanglement and quantum coherence.
The ground-state entanglement in a transverse spin-1/2 XX chain with a magnetization current is studied. By introducing a magnetization current to the system, a quantum phase transition to current-carrying phase may be presented with the variation of the driving field λ for the magnetic field h > 1; and the ground-state entanglement arises simultaneously at the critical point of quantum phase transition. In our model, the introduction of magnetization current may result in more entanglement between any two nearest-neighbour spins.
We investigate the emergence and properties of long-distance genuine multipartite entanglement, induced via three localized magnetic defects, in a one-dimensional transverse-field
XX spin-
1/2 chain. Using both analytical and numerical techniques, we determine the conditions for the existence of bound states localized at the defects. We find that the reduced density matrix (RDM) of the defects exhibits long-distance genuine multipartite entanglement (GME) across the whole range of the Hamiltonian parameter space, including regions where the two-qubit concurrence is zero. We quantify the entanglement by using numerical lower bounds for the GME concurrence, as well as by analytically deriving the GME concurrence in regions where the RDM is of rank 2. Our work provides insights into generating multipartite entanglement in many-body quantum systems via local control techniques.
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
N-qubit chain with nearest-neighbor
XXZ 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.
Abstract A way to characterize multipartite entanglement in pure states of a spin chain with sites and local dimension is by means of the Cayley hyperdeterminant. The latter quantity is a polynomial constructed with the components of the wave function which is invariant under local unitary transformation. For spin 1/2 chains (i.e. ) with and sites, the hyperdeterminant coincides with the concurrence and the tangle respectively. In this paper we consider spin chains with sites where the hyperdeterminant is a polynomial of degree 24 containing around terms. This huge object can be written in terms of more simple polynomials and of degrees 8 and 12 respectively. Correspondingly we compute , and the hyperdeterminant for eigenstates of the following spin chain Hamiltonians: the transverse Ising model, the XXZ Heisenberg model and the Haldane–Shastry model. Those invariants are also computed for random states, the ground states of random matrix Hamiltonians in the Wigner–Dyson Gaussian ensembles and the quadripartite entangled states defined by Verstraete et al in 2002. Finally, we propose a generalization of the hyperdeterminant to thermal density matrices. We observe how these polynomials are able to capture the phase transitions present in the models studied as well as a subclass of quadripartite entanglement present in the eigenstates.