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arXiv:quant-ph/0603114 · Physical Review Letters

General Entanglement Scaling Laws from Time Evolution

Jens Eisert, Tobias J. Osborne

Published Oct 12, 2006• 201 citations
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Abstract

We establish a general scaling law for the entanglement of a large class of ground states and dynamically evolving states of quantum spin chains: we show that the geometric entropy of a distinguished block saturates, and hence follows an entanglement-boundary law. These results apply to any ground state of a gapped model resulting from dynamics generated by a local Hamiltonian, as well as, dually, to states that are generated via a sudden quench of an interaction as recently studied in the case of dynamics of quantum phase transitions. We achieve these results by exploiting ideas from quantum information theory and tools provided by Lieb-Robinson bounds. We also show that there exist noncritical fermionic systems and equivalent spin chains with rapidly decaying interactions violating this entanglement-boundary law. Implications for the classical simulatability are outlined.

DOI
10.1103/physrevlett.97.150404

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We establish a general scaling law for the entanglement of a large class of ground states and dynamically evolving states of quantum spin chains: we show that the geometric entropy of a distinguished block saturates, and hence follows an entanglement-boundary law. These results apply to any ground state of a gapped model resulting from dynamics generated by a local Hamiltonian, as well as, dually, to states that are generated via a sudden quench of an interaction as recently studied in the case of dynamics of quantum phase transitions. We achieve these results by exploiting ideas from quantum information theory and tools provided by Lieb-Robinson bounds. We also show that there exist noncritical fermionic systems and equivalent spin chains with rapidly decaying interactions violating this entanglement-boundary law. Implications for the classical simulatability are outlined.
By means of free fermionic techniques we study the time evolution of the entanglement entropy S(t)\mathcal{S}(t)S(t) of a block of spins in the random transverse-field Ising chain after a sudden change of the parameters of the Hamiltonian. We consider global quenches, when the parameters are modified uniformly in space, as well as local quenches, when two disconnected blocks are suddenly joined together. For a noncritical final state, the dynamical entanglement entropy is found to approach a finite limiting value for both types of quenches. If the quench is performed to the critical state, the entropy grows for an infinite block as $\mathcal{S}(t)\ensuremath{\sim}\mathrm{ln}\mathrm{ln}t$. This type of ultraslow increase is explained through the strong disorder renormalization group method.
We prove that the entanglement entropy of any state evolved under an arbitrary 1/r^{α} long-range-interacting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α>D+1. We also prove that for any α>2D+2, the ground state of such a Hamiltonian satisfies the entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known to satisfy the area law. These results significantly generalize their existing counterparts for short-range interacting systems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presence of long-range interactions.
We investigate the entanglement spreading in the anisotropic spin-1/21/21/2 Heisenberg (XXZ)(XXZ)(XXZ) chain after a geometric quench. This corresponds to a sudden change of the geometry of the chain or, in the equivalent language of interacting fermions confined in a box trap, to a sudden increase of the trap size. The entanglement dynamics after the quench is associated with the ballistic propagation of a magnetization wave front. At the free fermion point (XXXXXX chain), the von Neumann entropy SA{S}_{A}SA​ exhibits several intriguing dynamical regimes. Specifically, at short times a logarithmic increase is observed, similar to local quenches. This is accurately described by an analytic formula that we derive from heuristic arguments. At intermediate times partial revivals of the short-time dynamics are superposed with a power-law increase ${S}_{A}\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}<1$. Finally, at very long times a steady state develops with constant entanglement entropy, apart from oscillations. As expected, since the model is integrable, we find that the steady state is nonthermal, although it exhibits extensive entanglement entropy. We also investigate the entanglement dynamics after the quench from a finite to the infinite chain (sudden expansion). While at long times the entanglement vanishes, we demonstrate that its relaxation dynamics exhibits a number of scaling properties. Finally, we discuss the short-time entanglement dynamics in the XXZXXZXXZ chain in the gapless phase. The same formula that describes the time dependence for the XXXXXX chain remains valid in the whole gapless phase.
The nature of entanglement in many-body systems is a focus of intense research with the observation that entanglement holds interesting information about quantum correlations in large systems and their relation to phase transitions. In particular, it is well known that although generic, many-body states have large, extensive entropy, ground states of reasonable local Hamiltonians carry much smaller entropy, often associated with the boundary length through the so-called area law. Here we introduce a continuous family of frustration-free Hamiltonians with exactly solvable ground states and uncover a remarkable quantum phase transition whereby the entanglement scaling changes from area law into extensively large entropy. This transition shows that entanglement in many-body systems may be enhanced under special circumstances with a potential for generating "useful" entanglement for the purpose of quantum computing and that the full implications of locality and its restrictions on possible ground states may hold further surprises.
Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quantum evolution. The change can be made by spatially local perturbations (local quench) or globally switching to a completely different Hamiltonian (global quench). This thesis investigates the post-quench non-equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench. We begin with two local quench examples. In Chap. 2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We find that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator -- a value representing its strength. In Chap. 3, we study the evolution after connecting two different one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory. In the global quench scenario, the change of the Hamiltonian affects all parts of the system. In this thesis, we focus on the global chaotic quench driven by generic non-integrable Hamiltonians. In Chap. 4, we propose to use the operator entanglement entropy of the unitary operator as a probe. Its fast linear entanglement production is sharply contrasted to the slow logarithmic spreading of the many-body localized system. The entanglement saturation suggests that the evolution operator in the long time can be modeled by a random unitary matrix. In Chap. 5, we construct a random tensor network which consists of random unitary matrices connected locally to model chaotic evolution with local interactions. We find that the entanglement dynamics is mapped to the statistical mechanics of interacting random walks. This appealing emergent picture allows us to understand the universal linear growth as well as the fluctuations of entanglement in a chaotic quench.