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: $D\le D_0+(\alpha v_{\mathrm{LR}}\tau +\beta \xi )v_C$. Here $v_{\mathrm{LR}}$ is the Lieb-Robinson velocity, $v_C$ is a velocity defined by the current operator, $\tau$ is the decoherence time, $\xi$ is the range of interactions, $D_0$ is a decoherence-induced microscopic 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 ($D_0$, $v_{\mathrm{LR}}$, $v_C$, $\xi$) or else determined from independent local nontransport 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.

Figure 1

Diffusivity $D$ of the dissipative $XXZ$ model versus dephasing strength $c$, with anisotropies $\mathrm{\Delta }=0.5$, 1.0, 1.5. The asymptotic behavior $D\sim 1/c$ is also shown. Operator spaces are truncated in numerics so that only Pauli string operators of length at most $n=7$ are kept. Finite-size effects are strong for small $c$ and indicated by the shaded region, which is estimated from truncations with $n=6$, 8.