Locality Bound for Dissipative Quantum Transport

Abstract

We prove an upper bound on the diffusivity of a general local and translation invariant quantum Markovian spin system: $D\le D_0+\left(\alpha v_{\text{LR}}\tau +\beta \xi \right)v_{\text{C}}$. Here $v_{\text{LR}}$ is the Lieb-Robinson velocity, $v_{\text{C}}$ is a velocity defined by the current operator, $\tau$ is the decoherence time, $\xi$ is the range of interactions, $D_0$ is a microscopically determined diffusivity and $\alpha$ and $\beta$ are precisely defined dimensionless coefficients. The bound constrains quantum transport by quantities that can either be obtained from the microscopic interactions ($D_0,v_{\text{LR}},v_{\text{C}},\xi$) or else determined from independent local non-transport measurements ($\tau ,\alpha ,\beta$). We illustrate the general result with the case of a spin half XXZ chain with on-site dephasing. Our result generalizes the Lieb-Robinson bound to constrain the sub-ballistic diffusion of conserved densities in a dissipative setting.

5 pages, 1 figure, revised for PRL