Entanglement dynamics in quantum many-body systems

The dynamics of entanglement has recently been realized as a useful probe in studying ergodicity and its breakdown in quantum many-body systems. In this paper, we study theoretically the growth of entanglement in quantum many-body systems and propose a method to measure it experimentally. We show that entanglement growth is related to the spreading of local operators in real space. We present a simple toy model for ergodic systems in which linear spreading of operators results in a universal, linear-in-time growth of entanglement for initial product states, in contrast with the logarithmic growth of entanglement in many-body localized (MBL) systems. Furthermore, we show that entanglement growth is directly related to the decay of the Loschmidt echo in a composite system comprised of several copies of the original system, in which connections are controlled by a quantum switch (two-level system). By measuring only the switch's dynamics, the growth of the Rényi entropies can be extracted. Our work provides a way of understanding entanglement dynamics in many-body systems and to directly measure its growth in time via a single local measurement.

Figure 1

Setup to measure the growth of the $n\mathrm{th}$ Rényi entropy $S_n\left(t\right)$. Here $n=3$ and we have shown it for a $d=1$ chain. There is a quantum switch in the middle of the setup which governs tunneling between different subsystems. (a) When the quantum switch is in the state $|\uparrow \rangle$, tunneling between $A_i$ and $B_i$ is allowed while tunneling between $A_i$ and $B_{i+1}$ is prohibited. (b) When the quantum switch is in the state $|\downarrow \rangle$, the allowed and prohibited tunnelings are swapped. A composite state in an $n$-copy product state of the chains and in a superposition of $|\uparrow \rangle$ and $|\downarrow \rangle$ of the quantum switch will have two parts evolving in time differently according to the quantum switch; measuring ${\sigma }^x\left(t\right)$ of the quantum switch gives the Loschmidt echo and hence the $n\mathrm{th}$ Rényi entropy.

Figure 2

(a) The support of a typical basis operator ${\mathcal{O}}_A$. Here $\mathrm{\Lambda }$ is a $d=2$ square lattice, and the subregion $A$ is a circle of radius $r_A$ demarcated by the blue circle. Black sites represent the presence of a nontrivial operator, ${\sigma }_i^x,{\sigma }_i^y$, or ${\sigma }_i^z$, while white sites represent the presence of an identity operator ${\mathbb{I}}_i$. One can see clusters of white sites within the circle. A site within a cluster is $\sim O\left(1\right)$ site away from a black site. (b) Under time evolution, each black site spreads with velocity $v$, and so the clusters quickly get filled up (red sites) in time $vt\sim O\left(1\right)$, and simultaneously the operator also grows in physical size to become a ball of radius $r_A+vt$.