A short review on entanglement in quantum spin systems

TL;DR

This review surveys how entanglement entropy encodes quantum correlations and critical behavior in one-dimensional spin systems. By combining explicit XX-model calculations with conformal-field-theory insights, it shows universal logarithmic scaling at criticality and area-law tendencies away from critical points, including extensions to XY, XXZ, and disordered or collective models. It also connects entanglement dynamics under quenches to propagation bounds and explores how entanglement constrains quantum computation and motivates efficient classical simulations via tensor-network methods. Overall, entanglement stands as a fundamental criterion distinguishing tractable from intractable quantum many-body dynamics and computation, closely tied to central charges and area-law violations across diverse models.

Abstract

We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation.

Figures (6)

• Figure 1: The two terms of $\Lambda_k$, Eq. (\ref{['Eq:Lambda_k']}), are plotted for the particular case $\lambda=1$. We realize that if $2\cos \left(\frac{2\pi k}{N}\right)>\lambda$, $\Lambda_k <0$ while if $2\cos \left(\frac{2\pi k}{N}\right)<\lambda$, $\Lambda_k >0$.
• Figure 2: Entropy of the reduced density matrix of $L$ spins for the XX model in the limit $N\to\infty$, for two different values of the external magnetic field $\lambda$. The maximum entropy is reached when there is no applied external field ($\lambda=0$). The entropy decreases while the magnetic field increases until $\lambda=2$ when the system reaches the ferromagnetic limit and the ground state is a product state in the spin basis.
• Figure 3: Entropy of entanglement is shown to decrease monotonically along the RG trajectory that takes the external magnetic field $\lambda$ away from its critical value $\lambda^*=1$. Towards the left the flow takes the system to a GHZ-line state whereas, towards the right, the system is a product state.
• Figure 4: Schematic representation of the dynamics of block entropy. Entangled particles are emitted from the region $A$, they will contribute to the block entropy as long as one of the two particles ends in the region $B$ [from CC05].
• Figure 5: Structure of the quantum circuit performing the exact diagonalization of the XY Hamiltonian for 8 sites. The circuit follows the structure of a Bogoliubov transformation followed by a fast Fourier transform. Three types of gates are involved: type-B (responsible for the Bogoliubov transformation and depending on the external magnetic field $\lambda$ and the anisotropy parameter $\gamma$), type-fSWAP (depicted as crosses and necessary to implement the anti-commuting properties of fermions) and type-F (gates associated to the fast Fourier transform). Some initial gates have been eliminated since they only amount to some reordering of initial qubits [from VCL08].