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International Journal of Quantum Information

ENTANGLEMENT DYNAMICS IN QUANTUM SPIN CHAINS

MŁOSZ MICHALSKI

Published Jul 1, 2008• 1 citation
Publisher

Abstract

We study patterns of temporal evolution, creation and transport, of entanglement in quantum spin chains. The model used is an isotropic Heisenberg chain with periodic boundary conditions, and we obtain in analytic form the time dependence of concurrence and negativity of various pairs of spins in the system. The objective of the present study is to assess the usefulness of various averaged entanglement measures in the characterization of entangling properties of quantum evolutions.

DOI
10.1142/s021974990800402x

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We study patterns of temporal evolution, creation and transport, of entanglement in quantum spin chains. The model used is an isotropic Heisenberg chain with periodic boundary conditions, and we obtain in analytic form the time dependence of concurrence and negativity of various pairs of spins in the system. The objective of the present study is to assess the usefulness of various averaged entanglement measures in the characterization of entangling properties of quantum evolutions.
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We study the dynamics of a Heisenberg XYXYXY spin chain with an unknown state coded into one qubit or a pair of entangled qubits, with the rest of the spins being in a polarized state. The time evolution involves magnon excitations, and through them the entanglement is transported across the channel. For a large number of qubits, explicit formulas for the concurrences, measures for two-qubit entanglements, and the fidelity for recovering the state some distance away are calculated as functions of time. Initial states with an entangled pair of qubits show better fidelity, which takes its first maximum value at earlier times, compared to initial states with no entangled pair. In particular initial states with a pair of qubits in an unknown state $\ensuremath{\alpha}\ensuremath{\uparrow}\ensuremath{\uparrow}+\ensuremath{\beta}\ensuremath{\downarrow}\ensuremath{\downarrow}$ are best suited for quantum-state transport.
The phenomenon of quantum entanglement has a very important role in quantum mechanics. Particularly, the quantum spin chain provides a platform for theoretical and experimental investigation of many-body entanglement. In this paper, we investigate time evolution of entanglement in a four-qubit anisotropic Heisenberg XXZ chain with nearest neighboring (NN), the next nearest neighboring (NNN), and the Dzialoshinskii-Moriya (DM) interactions. Calculations of the entanglement evolution of the Werner state carried out in terms of concurrence for selected ranges of control parameters such as DM interaction, frustration, etc. The results show that for the Werner state, DM interaction and the frustration parameters play important roles. Furthermore, results show that the time evolution of the Werner state entanglement may be useful to capture the quantum phase transitions in quantum magnetic systems.
Entanglement sharing among pairs of spins in Heisenberg antiferromagnets is investigated using the concurrence measure. For a nondegenerate S=0S=0S=0 ground state, a simple formula relates the concurrence to the diagonal correlation function. The concurrence length is seen to be extremely short. Various finite clusters are studied numerically, to see the trend in two dimensions. It is argued that the concurrences vanish between pairs of spins which are not nearest neighbors for the linear-chain and square-lattice antiferromagnet ground states. For the triangular lattice and Kagome' lattice, nearest-neighbor concurrences also vanish. The concurrences in the maximal-spin states are explicitly calculated, where the concurrence averaged over all pairs is larger than the S=0S=0S=0 states.
By using the concept of negativity, we study entanglement in spin-one Heisenberg chains. Both the bilinear chain and the bilinear-biquadratic chain are considered. Due to the SU(2) symmetry, the negativity can be determined by two correlators, which greatly facilitate the study of entanglement properties. Analytical results of negativity are obtained in the bilinear model up to four spins and the two-spin bilinear-biquadratic model, and numerical results of negativity are presented. We determine the threshold temperature before which the thermal state is doomed to be entangled.
Entanglement has been extensively used to characterize the structure of strongly correlated many-body systems. Most of these analyses focus on either spatial properties of entanglement or its temporal behavior. Negativity, as an entanglement measure, quantifies entanglement between different noncomplementary blocks of a many-body system. Here, we consider a combined spatial-temporal analysis of entanglement negativity in a strongly correlated many-body system to characterize complex formation of correlations through nonequilibrium dynamics of such systems. A bond defect is introduced through a local quench at one of the boundaries of a uniform Heisenberg spin chain. Using negativity and entanglement entropy, computed by the time-dependent density-matrix renormalization group, we analyze the extension of entanglement in the model as a function of time. We find that an entanglement cloud is formed, detached from the boundary spin and composed of spins with which it is highly entangled. The cloud travels ballistically in the chain until it reaches the other end where it reflects back and the cycle repeats. The revival dynamics exhibits an intriguing contraction (expansion) of the cloud as it moves away from (towards) the boundary spin.