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We consider one-dimensional translationally invariant quantum spin (orfermionic) lattices and prove a Mazur-type inequality bounding thetime-averaged thermodynamic limit of a finite-temperature expectation of aspatio-temporal autocorrelation function of a local observable in terms ofquasi-local conservation laws with open boundary conditions. Namely, thecommutator between the Hamiltonian and the conservation law of a finite chainmay result in boundary terms only. No reference to techniques used in Suzuki'sproof of Mazur bound is made (which strictly applies only to finite-sizesystems with exact conservation laws), but Lieb-Robinson bounds and exponentialclustering theorems of quasi-local C^* quantum spin algebras are invokedinstead. Our result has an important application in the transport theory ofquantum spin chains, in particular it provides rigorous non-trivial examples ofpositive finite-temperature spin Drude weight in the anisotropic Heisenberg XXZspin 1/2 chain [Phys. Rev. Lett. 106, 217206 (2011)].

Thermodyamic bounds on Drude weights in terms of almost-conserved quantities