Dynamics of Entanglement in One-Dimensional Spin Systems

TL;DR

The paper investigates how quantum entanglement propagates in exactly solvable one-dimensional spin-1/2 Ising-type chains by injecting Bell states into a fully polarized background or into the ground state. Using the Jordan-Wigner transformation and Pfaffian correlation-function techniques, it derives time-dependent correlators and computes entanglement measures such as concurrence and tangle. In the isotropic case, entanglement remains purely pairwise and travels with velocity λ, while anisotropy can induce instantaneous vacuum entanglement and lead to complex entanglement-sharing dynamics; the CKW conjecture is satisfied with zero residual entanglement in the isotropic regime. The results reveal how anisotropy and proximity to the quantum critical point modify entanglement transport, with implications for quantum information tasks in spin networks and for understanding entanglement sharing in many-body systems.

Abstract

We study the dynamics of quantum correlations in a class of exactly solvable Ising-type models. We analyze in particular the time evolution of initial Bell states created in a fully polarized background and on the ground state. We find that the pairwise entanglement propagates with a velocity proportional to the reduced interaction for all the four Bell states. Singlet-like states are favored during the propagation, in the sense that triplet-like states change their character during the propagation under certain circumstances. Characteristic for the anisotropic models is the instantaneous creation of pairwise entanglement from a fully polarized state; furthermore, the propagation of pairwise entanglement is suppressed in favor of a creation of different types of entanglement. The ``entanglement wave'' evolving from a Bell state on the ground state turns out to be very localized in space-time. Further support to a recently formulated conjecture on entanglement sharing is given.

Figures (21)

• Figure 1: Concurrence $C_{i,i+x}$ as a function of time and of the distance between the initially entangled sites. The initial state is a singlet ($\varphi=\pi$ in the initial condition) involving the very same sites $i$ and $i+x$.
• Figure 2: Concurrence between sites at a distance $x$ (namely, $n=i$, and $n=i+x$ in Eq. \ref{['concnm']}), for the case of an initial $0$-triplet state ($\varphi=0$) shared by two nearest neighbor sites, $i=0, j=1$. The various plots correspond to $x=1, \ldots, 8$.
• Figure 3: Concurrence between sites $n=-x, m=x$, symmetrically displaced from their initial position $i=-1$ and $j=1$ ($\varphi=\pi$).
• Figure 4: Summed concurrences for an initially entangled site ($n=0$, full line) and an initially unentangled site ($n=10$, dashed line). The initial state is a singlet ($\varphi=\pi$) created on sites $i=0$ and $j=1$.
• Figure 5: Entropy $S^{(2)}_{(-x,x+1)}(\tau)$ for pairs of sites symmetrically displaced with respect to the initial singlet position at $(i,j)= (0,1)$. This is for $\varphi = \pi$.