Locality Bound for Dissipative Quantum Transport

We prove an upper bound on the diffusivity of a dissipative, local, and translation invariant quantum Markovian spin system: D≤D_{0}+(αv_{LR}τ+βξ)v_{C}. Here v_{LR} is the Lieb-Robinson velocity, v_{C} is a velocity defined by the current operator, τ is the decoherence time, ξ is the range of interactions, D_{0} is a decoherence-induced microscopic diffusivity, and α and β are precisely defined dimensionless coefficients. The bound constrains quantum transport by quantities that can either be obtained from the microscopic interactions (D_{0}, v_{LR}, v_{C}, ξ) or else determined from independent local nontransport measurements (τ, α, β). 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.