Entanglement versus Correlations in Spin Systems

We consider pure quantum states of $N\gg 1$ spins or qubits and study the average entanglement that can be localized between two separated spins by performing local measurements on the other individual spins. We show that all classical correlation functions provide lower bounds to this localizable entanglement, which follows from the observation that classical correlations can always be increased by doing appropriate local measurements on the other qubits. We analyze the localizable entanglement in familiar spin systems and illustrate the results on the hand of the Ising spin model, in which we observe characteristic features for a quantum phase transition such as a diverging entanglement length.

Figure 1

Localizable entanglement $E_{ij}$ and correlation function $Q_{xx}^{ij}$ (solid) and upper bound (dashed) as a function of the coupling parameter $\lambda$ for the infinite Ising chain with spin distance $n=1$ (left) and $n=10$ (right). Observe the big gap that can occur between the upper bound (allowing global measurements) and the LE (local measurements) for large spin distance.

Figure 2

Correlation function $Q_{xx}^{i,i+4}$ (solid), localizable entanglement $E_{i,i+4}$ (diamonds), and its upper bound (dashed) as a function of the coupling parameter $\lambda$ for the ground state of the finite Ising chain ($N=14$) in the case of broken parity symmetry by a small perturbating magnetic field in the $x$ direction.