We prove an upper bound on the diffusivity of a general local and translation invariant quantum Markovian spin system: $D leq D_0 + left(alpha , v_text{LR} tau + beta , xi right) v_text{C}$. 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.
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