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The Lieb-Robinson bound sets a theoretical upper limit on the speed at whichinformation can propagate in non-relativistic quantum spin networks. In itsoriginal version, it results in an exponentially exploding function of theevolution time, which is partially mitigated by an exponentially decreasingterm that instead depends upon the distance covered by the signal (the ratiobetween the two exponents effectively defining an upper bound on thepropagation speed). In the present paper, by properly accounting for the freeparameters of the model, we show how to turn this construction into a strongerinequality where the upper limit only scales polynomially with respect to theevolution time. Our analysis applies to any chosen topology of the network, aslong as the range of the associated interaction is explicitly finite. For thespecial case of linear spin networks we present also an alternative derivationbased on a perturbative expansion approach which improves the previousinequality. In the same context we also establish a lower bound to the speed ofthe information spread which yields a non trivial result at least in the limitof small propagation times.

Time-Polynomial Lieb-Robinson bounds for finite-range spin-network models